Proof for Validation for‚ Construction of a Triangle with Given Perimeter and Two Base Angles

Q.

A student constructed a triangle with the known conditions to him being the perimeter of the triangle and both the base angles. The steps of construction he used are as follows:

1. Draw a line segment, say XY equal to AB + BC + AC

2. Make angles LXY equal to $\angle$ B and MYX equal to $\angle$ C

3. Bisect $\angle$ LXY and $\angle$ MXY. Let these bisectors intersect each other at A.

4. Draw perpendicular bisectors PQ of AX and RS of AY

5. Let PQ intersect XY at B and RS intersect XY at C. Join AB and AC.

6. The triangle ABC is thus formed.

How many isosceles triangles are there in this figure?

1. 0

2. 1

3. 2

4. 4

Q. Will the lengths of line segment $AB$ and line segment $BC$ make the length of line segment $AC$ in figure?