68 Chapter 6

Of course, we may also immediately deduce the following corollary.

“Corollary” 48. Every triangle is equilateral.

“Proof.” The proof of proposition 47 proceeded from an arbitrary vertex A of triangle

ABC, and so the argument actually shows that each pair of adjacent sides is congruent.

So it is equilateral.

P

R

Q

S

B

A

C

Perhaps someone might criticize the proof we

gave for proposition 47 by saying that we do not

necessarily know that the angle bisector at A inter-

sects BC on that side of the midpoint P. Perhaps the

intersection is on the other side, as in the diagram

here. This would cause the point Q to be exterior to

the triangle.

But the argument works just as easily for this case.

Namely, we again let Q be the intersection of the

angle bisector at ∠A with the perpendicular bisec-

tor of BC at midpoint P, and again drop the per-

pendiculars from Q to R and S . Again, we get

BQ  CQ by the Pythagorean theorem, using the

green triangles. And again, we get ARQ  AS Q

since these are similar triangles with the same hypotenuse. So again, we conclude

that BQR  CQS by the hypotenuse-leg congruence theorem. So we deduce that

AB  AR − BR  AS − CS  AC, and so the triangle is isosceles.

Mathematical Habits

Use insightful diagrams and figures. Whenever possible, augment your mathemat-

ical arguments with graphical aids. Convey your mathematical ideas visually. Take

the time to design informative pictures or diagrams. Use these graphical elements to

illustrate and clarify your main argument, but not to replace it.

Recognize the limitations of figures and diagrams. When they do not or cannot

represent the fully general case, figures and diagrams can sometimes lead to unwar-

ranted conclusions, which hold for that particular case but not generally. So take care

to consider whether your figures or diagrams might be suggesting particular rather

than general features.