Rhombus ADEF is inscribed into a triangle ABC so that they share angle A and the vertex E lies on the side BC . What is the length of the side of the rhombus if AB=c, and AC=b.

Question

Rhombus ADEF is inscribed into a triangle ABC so that they share angle A and the vertex E lies on the side 

BC

. What is the length of the side of the rhombus if AB=c, and AC=b.

Answer 1

Sam's answer involves trig and makes an assumption about angles.

However, no trig is necessary as follows, but it is necessary to draw a figure.

ΔBDE is similar to ΔEFC (and I will leave it to you to confirm the truth of that statement)

Then BD/EF = DE/CF because they are corresponding parts of similar triangles

Let s = side of the rhombus

Substituting (c-s)/s = s/(b-s) which means s = bc/(b+c)

Answer 2

We are talking trig. In this case angle A=γ; angle B=β; angle C=α.

Side AB=a; side AC=b; side BC=c.

In this case side a=side b.

You want side c. So: a^2+b^2-2ab*cosγ=c^2.