Although we have encountered many surprising relationships throughout the first five chapters, we find that there are some theorems that are particularly noteworthy, not only because they make our investigation of triangle properties much more efficient, but also because they expose many additional triangle properties. Each of these seems to provide a very surprising relationship in triangles and should serve as a springboard for further investigations.

MENELAUS'S THEOREM

Earlier we encountered the powerful relationship discovered by Giovanni Ceva that gave us concurrency from points on the sides of a triangle that determined equal products of alternate segments. Analogous to that relationship is one that Ceva discovered to have been developed in about 100 CE by Menelaus of Alexandria (70–140 CE), who established that the equal products of alternate segments on the sides of a triangle determine collinear points, as you can see from the following statement of Menelaus's theorem:

If three points, X, Y, and Z, are located where each of two of these points are on two different sides of triangle ABC, and the third point is on the extension of the third side of the triangle such that AZ · BX · CY = AY · BZ ⋅ CX, then the three points X, Y, and Z are collinear.

This can be seen in figures 6-1a and 6-1b.

The proof of this theorem can be found in the appendix. The converse of this theorem is also true. Namely, if the three points are collinear, with one on each side of a triangle (or extension), then the products of the alternate segments of the sides of the triangle are equal. We will use Menelaus's theorem as we further explore lines and points of a triangle—more specifically when we consider points on a triangle that are collinear.

SIMSON'S THEOREM

When considering collinear points involving triangles, the famous Simson theorem must be acknowledged. One of the great injustices in the history of mathematics involves this theorem. It was originally published by William Wallace (1768–1843) in Thomas Leybourn's Mathematical Repository (1799–1800), which through careless misquotes has been attributed to Robert Simson (1687–1768), the famous English interpreter of Euclid's Elements, whose book—as we mentioned earlier—has been the basis for the study of geometry in the English-speaking world and more specifically has greatly influenced the American high-school geometry course. To conform to the norm, we shall use the popular reference Simson's theorem throughout this book.

Simson's theorem states that the feet of the perpendiculars drawn from any point on the circumscribed circle of a triangle to the sides of the triangle are collinear. This is shown in figure 6-2, where point P is any point on the circumscribed circle of triangle ABC. We then draw PY AC at Y, PZ AB at Z, and PX BC at X. According to Simson's theorem, points X, Y, and Z are collinear. This line is usually referred to as the Simson line of P with respect to triangle ABC. It should be noted that the converse of this theorem is also true.

Before we discuss a number of aspects of this famous line, we will provide a proof of the theorem. Although Menelaus's theorem can be used to prove Simson's theorem (see appendix), we will provide a more elementary proof here.

In figure 6-3a, we will draw PA, PB, and PC. Since ∠PYA and ∠PZA are right angles, they are supplementary, thus establishing quadrilateral PZAY as cyclic (figure 6-3b shows the concyclic points).

(Recall that a cyclic quadrilateral is one that can be inscribed in a circle.)

Therefore, ∠PYZ = ∠PAZ.

(I)

Similarly, since ∠PYC is supplementary to ∠PXC, quadrilateral PXCY is cyclic, and

∠PYX = ∠PCB.

(II)

However, quadrilateral PACB is also cyclic, since it is inscribed in the given circumscribed circle, and therefore

∠PAZ = ∠PCB.

(III)

From equations (I), (II), and (III), we can establish that ∠PYZ = ∠PYX, and thus points X, Y, and Z are collinear.

 

A curiosity of the Simson line can be seen when we generate the Simson line from the intersection point on the circumscribed circle of the extension of one of the triangle's altitudes. This Simson line is parallel to the tangent at the vertex from which this altitude emanates. For example, when, in figure 6-4, the altitude BD = h_b of triangle ABC meets the circumscribed circle at P (and at B), then the Simson line of triangle ABC with respect to P is parallel to the line tangent to the circle at B.

This can be rather easily justified. We know that in figure 6-4 point D is one of the points on the Simson line. We also have PX and PZ perpendicular, respectively, to sides BC and AB of triangle ABC. Therefore, points X, D, and Z determine the Simson line of P with respect to triangle ABC.

Next, we will draw PC. Consider quadrilateral PDCX, where ∠PDC = ∠PXC = 90°, thus making PDCX a cyclic quadrilateral. The two inscribed angles intercepting the same arc of the circumscribed circle of quadrilateral PDCX are equal.

Therefore, ∠DXC = ∠DPC.

(I)

However, in the circumscribed circle (with circumcenter O) of triangle ABC,

∠EBC = , and ∠DPC(=∠BPC) = .
Therefore, ∠EBC = ∠DPC.

(II)

 

From (I) and (II), by transitivity, ∠DXC = ∠EBC, which are then alternate-interior angles, making the Simson line XDZ parallel to tangent line EB.

Another Simson-line curiosity is that if we have two Simson lines generated for the same triangle by two distinct points of the circumscribed circle, the angle formed by the two Simson lines is one-half the measure of the (larger) arc they intercept on the circle.

In figure 6-5, the two Simson lines YZX and UVW generated by the points P and Q, respectively, intersect the circle-determining arc . The angle, ∠MTN, formed by the two Simson lines has one-half the measure of the intercepted arc .

We also have another interesting property of the Simson line. A Simson line can be shown to bisect the line that joins the orthocenter with the generator point of the Simson line. We can see this in figure 6-6, where point P is used to generate the Simson line XZY of triangle ABC. The line PH, joining the orthocenter, H, of the triangle with point P, is bisected by the Simson line at point M, or PM = HM.

Using dynamic geometry software—such as the Geometer's Sketchpad^® or GeoGebra^®—the Simson line can be shown to generate a famous geometric shape called the Steiner deltoid or Steiner's hypocycloid, named after its founder, Jakob Steiner (1796–1863), who presented it in 1856.¹ When the point P moves around the circumscribed circle of triangle ABC, the Simson line generates the Steiner deltoid. (See figure 6-7.) The Simson lines are tangent to the newly formed deltoid.

The Steiner deltoid is tangent at three points to the nine-point circle of this triangle, which we will present later in this chapter. Its circumscribed circle is the Steiner circle, and its inscribed circle is the nine-point circle. The Steiner circle is shown in figure 6-7. The center point of the Steiner circle is N, which is also the center point of the nine-point circle, its radius is R.

The Simson line has many other gems that we shall leave to the reader to discover.²

THE EULER LINE

We have already seen a fine collection of significant points of a triangle, such as the orthocenter, the incenter, the centroid, the circumcenter, the Gergonne point, the Nagel point, the middle point, the Fermat point, the Brocard points, the Miquel point, and symmedian point. However, in high school we already encountered some of these points, such as the orthocenter H, the centroid G, and the circumcenter O. These points enable us to reveal a surprising result: They are collinear! In 1763,3 the famous Swiss mathematician Leonhard Euler (1707–1783) discovered that these points were truly collinear—as long as the triangle is not equilateral, then they would coincide. (See figure 6-8.) This line of collinear points is therefore called the Euler line, e.

Dynamic geometry software quickly shows us that the three points are collinear. But this is merely a demonstration. To establish this relationship's truth, we would have to prove the collinearity. This was then Euler's task as well, since he used rather primitive tools to discover this collinearity and then had to develop a proof. As we search for a method of proof, we realize that we cannot use Menelaus's theorem, since the points in question do not lie on the sides of the triangle.

Furthermore, we can show that the centroid, G, of a triangle trisects the segment HO, the segment from the orthocenter H to the circumscribed center O. (See figure 6-9.)

To prove this collinearity, consider in triangle ABC, the line GO (figure 6-10), formed by the centroid (G) and the center of the circumscribed circle (O), where we will then place the point P so that . We will now set out to show that point P is actually the orthocenter, point H.

Recall that in triangle ABC the line segment AM_a is a median and is divided by the centroid so that . Since ΔOGM_a ~ΔPGA (two pairs of corresponding sides are in proportion and the included angles are equal), we have OM_a AP. Since the line from the center of a circle to the midpoint of a chord is perpendicular to the chord, OM_a BC. Therefore, APD is an altitude of the triangle, since AP BC. That means that point P is on the altitude AH_a (D = H_a). We can repeat this for the other altitudes and then establish that the point P coincides with the orthocenter H.4

We should also note that the center of the inscribed circle of a triangle (I), along with the centroid (G), and the Nagel point (N) are also collinear.

THE NINE-POINT CIRCLE

We know that any three noncollinear points lie on a unique circle. Some quadrilaterals have all four of their vertices also on a unique circle. These quadrilaterals we called cyclic quadrilaterals. A general parallelogram—other than a square or a rectangle—does not have all of its vertices on a unique circle. For a quadrilateral to have all of its vertices on a circle, the opposite angles must be supplementary, as is the case, for example, with an isosceles trapezoid. However, to find more than four points that lie on the same circle has been a long-standing challenge for mathematicians.

In 1765, Leonhard Euler showed that there are six points of a triangle that lie on a unique circle, namely, the midpoints of the sides, and the feet of the altitudes. Yet not until 1820, when a paper5 published by the French mathematicians Charles Julien Brianchon (1783–1864) and Jean-Victor Poncelet (1788–1867) appeared, were an additional three points of a triangle added to Euler's circle of six points. These new points were the midpoints of the segments from the orthocenter to the vertices. This paper contained the first complete proof that, in fact, these nine points all lie on the same circle and therefore gave the circle a name—the nine-point circle.

The German mathematician Karl Wilhelm Feuerbach (1800–1834) has much of his fame resting on a paper he published in 1822, where he stated that “the circle which passes through the feet of the altitudes of a triangle touches all four of the circles which are tangent to the three sides of the triangle” (Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreicks). Here he was referring to the triangle's inscribed circle and the three escribed circles—those externally tangent to each of the sides of the triangle. (See figures 6-11 and 6-12.) As a result of this work, the theorem is referred to as the Feuerbach theorem and the nine-point circle is also sometimes called the Feuerbach circle.

The thirteen points on the nine-point circle, c_N, with the center N are as follows (figure 6-12):

Nine-point Circle, c_N (Euler; Brianchon; Poncelet):
M_a, M_b, M_c – midpoints of the sides of triangle ABC,
H_a, H_b, H_c – feet of the altitudes of triangle ABC,
E_a, E_b, E_c – the so-called Euler points of triangle ABC (these are the
      midpoints of the segments between the orthocenter H and the
      vertices of triangle ABC) (Feuerbach):

F_i – point of tangency with the inscribed circle (at I),
F_a – point of tangency with the escribed circle (at side a),
F_b – point of tangency with the escribed circle (at side b),
F_c – point of tangency with the escribed circle (at side c).

Roger A. Johnson (1890–1954) in his landmark book writes Feuerbach's famous theorem “is perhaps the most famous of all theorems of the triangle, aside from those known in ancient times.”6 A similar sentiment is shared by the American mathematician Howard Eves (1911–2004).⁷

Let us now begin to justify the placement of these nine points on one circle. We will begin with the first six points. Remember each of the sets of three points will automatically lie on a circle (since they are not collinear). We shall begin by showing that the three midpoints of the sides of the triangle and the foot of one of the altitudes will all lie on the same circle.

We begin by trying to establish four points on a circle. In figure 6-13, points M_a, M_b, and M_c are the midpoints of the three sides, BC, AC, and AB, respectively, of triangle ABC. We also have CH_c as an altitude of triangle ABC. Since M_aM_b is a midline of triangle ABC, M_aM_b AB. Therefore, quadrilateral M_aM_bM_cH_c is a trapezoid.Also,M_bM_c is a midline of triangle ABC, so that M_bM_c = BC. Since M_aH_c is the median to the hypotenuse of right triangle BCH_c, we have M_aH_c = BC. Therefore M_bM_c = M_aH_c and trapezoid M_aM_bM_cH_c is isosceles. You will recall that when the opposite angles of a quadrilateral are supplementary, as in the case of an isosceles trapezoid, the quadrilateral is cyclic. Therefore, quadrilateral M_aM_bM_cH_c is cyclic, and we now have established that these four points lie on the same circle.

To simplify matters, we redraw triangle ABC (figure 6-14), this time with altitude AH_a. Using the same argument as before, we find that quadrilateral M_aM_bM_cH_a is an isosceles trapezoid and is therefore cyclic. So we now have five points on one circle (i.e., points M_a, M_b, M_c, H_c, and H_a). By repeating the same argument for altitude BH_b, we can then state that points H_a, H_b, and H_c lie on the same circle as points M_a, M_b, and M_c.

In figure 6-15 we see the six points (M_a, M_b, M_c, H_a, H_b, and H_c) that Euler established as lying on one circle.

We will now embark on our quest to establish another three points on the circle that already has six established points on it. In figures 6-16a and 6-16b, we have H as the orthocenter (the point of intersection of the altitudes), and E_c is the midpoint of CH. (E_a, E_b, and E_c are the midpoints of the segments AH, BH, and CH—the so-called Euler points of triangle ABC.)

It is this point, E_c, that we want to establish on the circle of six points. The line M_bE_c, which is a midline of triangle ACH, is parallel to AH, or essentially to altitude AH_a. Since M_bM_c is a midline of triangle ABC, it follows that M_bM_c is parallel to BC. Therefore, since triangle AH_aC is a right triangle, and the sides of triangle E_cM_bM_c are parallel to the sides of triangle AH_aC, then ∠E_cM_bM_c is also a right angle. Thus, quadrilateral E_cM_bM_cH_c is cyclic (opposite angles are supplementary). This places point E_c on the circle determined by points M_b, M_c, and H_c. We now have seven points on the one circle.

We can repeat this procedure with point E_b, the midpoint of BH (see figures 6-17a and 6-17b). As before, ∠M_bM_aE_b is a right angle, as is ∠M_bH_bE_b. Therefore points M_b, H_b, E_b, and M_a are concyclic (opposite angles are supplementary). We now have E_b as an additional point on our circle, making it an eight-point circle.

To locate our final point on the circle, consider point E_a, the midpoint of AH. As we did earlier, we find ∠M_aM_bE_a to be a right angle, as is ∠M_aH_aE_a. Therefore, quadrilateral M_aH_aE_aM_b is cyclic and point E_a is on the same circle as points M_b, M_a, and H_a. We have shown, therefore, that nine specific points lie on this circle. (See figure 6-18.)

We have established that there are nine points on the same circle, called the nine-point circle, namely, the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments from the orthocenter to the vertices. With a bit of further exploration, we can find some rather novel properties of this circle, some of which we shall identify now.

SOME PROPERTIES RELATED TO THE NINE-POINT CIRCLE

As we refer to figure 6-19, we will show some rather unexpected properties that this already-unique nine-point circle provides:

• The center, N, of the nine-point circle of a triangle is the midpoint of the segment, HO, that is, the segment from the orthocenter to the center of the circumscribed circle. This means that the center, N, of the nine-point circle lies on the Euler line. (See appendix for proof.)
• The centroid of a triangle, G, trisects the Euler line segment, HO, the segment from the orthocenter to the center of the circumscribed circle: HG = 2 GO, or . (See p. 144, figure 6-9.)

  

• The nine-point circle of the triangle ABC is also the circumscribed circle for each of the following triangles:

  –Triangle M_aM_bM_c, the triangle formed by the midpoints of the sides of the original triangle.

  – Triangle H_aH_bH_c, the triangle formed by the feet of the altitudes.

  – The Euler triangle E_aE_bE_c.

• The length of the radius of the nine-point circle of a triangle is one-half the length of the radius of the circumscribed circle. This can be seen in figure 6-19, where NE_c = OC, and in this case also parallel to it, which helps justify the statement. Six of the points on the nine-point circle are the vertices of its diameter:

  M_aE_a = M_bE_b = M_cE_c (= OC). (To justify this, note that ∠M_aM_bE_a and ∠M_aH_aE_a are right angles.)

• All triangles inscribed in a given circle and having a common orthocenter also have the same nine-point circle.
• Tangents to the nine-point circle of a triangle at the midpoints of the sides of the triangle are parallel to the sides of the orthic triangle.

  (When we connect the feet of the altitudes, we form a special pedal triangle of the original triangle, a so-called orthic triangle H_aH_bH_c.)

  One such tangent (at point M_c) is shown in figure 6-20 to be parallel to side H_aH_b of the orthic triangle H_aH_bH_c.

  

• Tangents to the nine-point circle at the midpoints of the sides of the given triangle are parallel to the tangents to the circumscribed circle at the opposite vertices of the given triangle. In figure 6-20, the tangent at point C is parallel to the tangent at point M_c.
• The four triangles formed by the four points—the vertices of the triangle and the orthocenter—create an orthocentric system, each of whose triangles has the same orthic triangle and the same nine-point circle. (An orthocentric system is a set of four coplanar points, each of which is the orthocenter of the triangle formed by the remaining three points.)
• In figure 6-21, the four triangles forming an orthocentric system are:

  – ΔABC with the orthocenter H,

  – ΔAHC with the orthocenter B,

  – ΔBHC with the orthocenter A, and

  – ΔAHB with the orthocenter C.

  

  It is also interesting to note that the radii of the four circumscribed circles of these triangles are equal (OA = PB = QC = RA). Furthermore, if we take the centers of these circumscribed circles, we will find that they have the same nine-point circle as the original orthocentric system. These circles are shown in figure 6-22.

  

  We can make an interesting extension: The four centroids of the four triangles of an orthocentric system also form an orthocentric system whose nine-point circle is concentric with that of the nine-point circle of the original orthocentric system. We leave this relationship and others to be found here for the reader to discover.

• The nine-point circle c_N of a triangle ABC is tangent to the inscribed circle and the escribed circles of the triangle. This is shown in figure 6-23. (See also figures 6-11 and 6-12, where you will find the names of the thirteen points).

The last of these properties is—as we indicated earlier—one of the most famous properties of the nine-point circle and was first discovered (and proved) by Karl Wilhelm Feuerbach, a German mathematician, in 1822, and consequently bears his name. This property establishes a relationship between the nine-point circle and the inscribed circle and escribed circles of a triangle.

The point F_i of tangency of the nine-point circle c_N with the inscribed circle is the so-called Feuerbach point. Also, the triangle F_aF_bF_c is named after Feuerbach.

The justifications (or proofs) for some of these properties of the nine-point circle can be found in the appendix.

With this all-encompassing circle, we are about to bring our chapter on the various lines, points, and circles of a triangle to a close. We have tried to provide a view of the most dramatic and often-overlooked relationships among the primary components of plane geometry. There are—needless to say—many more to be found. This pleasure we leave to the reader. Yet one more relationship merits mention and is a fine place to leave the reader in wonderment.

MORLEY'S THEOREM

We will close this chapter on triangle parts with a very famous theorem that was first published in 1900 by Frank Morley (1860–1937) and is one of the more difficult theorems in geometry to prove. (A proof is offered in the appendix.) Yet its beauty lies in the simplicity of the statement: The intersections of the adjacent angle trisectors of any triangle always meet in three points determining an equilateral triangle. Figure 6-24 shows a number of differently shaped triangles with their trisectors drawn, and in each case, the intersections of the adjacent trisectors determine an equilateral triangle. We invite the reader to try this with a variety of differently shaped triangles to see that it holds true for all triangles. Using dynamic geometry software would provide a very dramatic appreciation for this amazing relationship.

Beyond Morley's wonderful discovery, you can discover eighteen equilateral triangles under the “Morley's trisector theorem” entry on Wikipedia (http://en.wikipedia.org/wiki/Morley%27s_trisector_theorem).

Not to be disappointed by a possible neglect of a concurrency, yes, we have one in this configuration as well. In figure 6-25, we notice that CD, AF, and BE are concurrent.

Concurrency, collinearity, tangency, parallelism, and perpendicularity, along with the many relationships of the various parts of a triangle provide a seemingly endless array of properties that seem to be secretly nested in the triangle. One can continue to endlessly pursue other triangle relationships; however, we feel that we have captured the essence of these delights here in this chapter. After all, we must leave some of these geometric gems for the reader to discover!8