PES is an isosceles triangle ,then there's a triangle inside it ,how does it become an isosceles triangle,the middle one?

If PES is an isosceles triangle, then the triangle inside it must also be an isosceles triangle. This is because the base angles of an isosceles triangle are equal, and if a triangle is inscribed inside another triangle with equal base angles, then the inner triangle will also have equal base angles, making it an isosceles triangle.

Let's assume PES is an isosceles triangle with PE = PS.

To create an isosceles triangle inside PES, draw a line segment QR parallel to PS such that QR lies between P and S.

Now, we have two triangles: PQR and QRS.

Since QR is parallel to PS, we have:

∠PQR = ∠PSE (alternate interior angles)

∠QRS = ∠PES (alternate interior angles)

And since PE = PS, we also have:

∠PEQ = ∠PSQ

Therefore, triangles PQR and QRS are similar by the Angle-Angle-Angle (AAA) similarity theorem.

This means that:

PQ/PS = QR/QS = PR/SQ

Since PS = PE, we can substitute PE for PS:

PQ/PE = QR/QS = PR/SQ

Because PQ = QR (since it is the same line segment), we have:

PE/PE = QR/QS = PR/SQ

Simplifying, we get:

1 = QR/QS = PR/SQ

Therefore, QR = QS and PR = SQ.

This means that triangle QRS is isosceles, with QR = QS.