Following statement posed to GPT - no answer!

Geometry problem: Semi-circle inside triangle: Triangle with known length sides a, b, c where a is the longest. Place inside the triangle a semi-circle with diameter on side a. What is radius of largest possible semi-circle in terms of side lengths? Position of diameter center along a?

"

This is a challenging problem. I will try to help you by using some facts from geometry.

12, which says that if a triangle is inscribed inside a circle, where one side of the triangle is the diameter of the circle, then the angle opposite to that side is a right angle.

So, if we draw a semi-circle with diameter on side

**a**, then the angle opposite to

**a** is a right angle. Let’s call this angle

**C**.

Now, we can use the

**Pythagorean theorem** to find the length of

**a** in terms of

**b** and

**c**. We have:

a2=b2+c2

3 to find the radius of the semi-circle in terms of

**a** and

**C**. We have:

fracasinC=2R

where

**R** is the radius of the semi-circle.

Since we know that

**C** is a right angle, we have:

sinC=1

So, we get:

a=2R

Substituting this into the Pythagorean theorem, we get:

(2R)2=b2+c2

Solving for

**R**, we get:

R=fracsqrtb2+c22

This is the radius of the largest possible semi-circle in terms of side lengths.

To find the position of the diameter center along

**a**, we can use similar triangles. We have:

fracRb=fraca−Rc

Substituting

**a = 2R**, we get:

fracRb=fracRc

Cross-multiplying, we get:

cR=bR

Dividing by

**R**, we get:

c=b

This means that the triangle is an isosceles right triangle, and the diameter center is at the midpoint of

**a**.

I hope this helps you with your problem. "

From Bing Chat