MathDB

In a triangle

ABC

, the bisector of the largest angle

\angle A

meets

BC

at point

D

. Let

E

and

F

be the feet of perpendiculars from

D

AC

and

AB

, respectively. Let

R

denote the ratio between the areas of triangles

DEB

and

DFC

. (a) Prove that, for every real number

r > 0

, one can construct a triangle ABC for which

R

is equal to

r

. (b) Prove that if

R

is irrational, then at least one side length of

\vartriangle ABC

is irrational. (c) Give an example of a triangle

ABC

with exactly two sides of irrational length, but with rational

R

Problem Statement