Math help - geometric proofs please help?

Question

I have 2 questions i just can not get. Please help me in any way you can, being as descriptive as you can while still speaking simply (please explain the terms you use.)

first, there is:

Use you knowledge of circle theorems to proove that you have produced an incirle and a circumcircle. THIS pretty much says proove that you have made a triangle inside a circle, and a circle inside a triangle. Now i know you can't see that picture, bu anyone know how i can proove it?

And there is also:

Prove that for all right angle triangles that the midpoint of the hypotenuse is always the circumcentre of the circumcircle.

Help please?

Answer 1

1.

The centre of the circumcircle has to be equidistant from all three vertices.

In triangle ABC, therefore, it is equidistant from A and B ...(1)

It is also equidistant from B and C ...(2)

All points satisfying (1) lie on the perpendicular bisector of AB. All points satisfying (2) lie on the perpendicular bisector of BC. Where these two lines meet is therefore the centre of the circumcircle.

If you bisect the internal angles instead of the sides, then you can apply a similar  argument for the incircle.

The centre of the incircle must be equidistant from AB and from BC. It must lie on the bisector of angle ABC.

It must also be equidistant from BC and from CA. It must lie on the bisector of angle BCA.

Therefore every triangle has both a circumcircle and an incircle.

2.

The angle in a semicircle is a right angle.

Therefore,  for a right angled triangle, the centre of the circumcircle must lie on the hypotenuse, making the hypotenuse a diameter.

The centre has to be equidistant from all three vertices, and must therefore be at the midpoint of the hypotenuse.