Given isosceles △ABC, choose a point P on base BC. Show that △ABP and △ACP have equal circumradii.

Question

Given isosceles △ABC, choose a point P on base BC. Show, without using the law of sines, that △ABP and △ACP have equal circumradii. (Hint. Draw the ray AP′ so that AP and AP′ are on opposite sides of AC, ∠BAP = ∠CAP′ and AP = AP′ . Show that ∠APC and AP′C are supplementary. Please draw pics in answers.

Answer 1

When you draw the required figure you will see that the 2 circumcircles each pass through the points A and P.

The arcs subtended by angles B and C are equiangular, i.e. they have the same central angle.

Then the triangles formed by the points A and P and each circumcenter are congruent so that the circumradii are equal.

At least in my opinion it is imperative that you draw a careful figure in order to see the relationships.

The figure must assume the result and then prove (as above) that the assumed result is correct.