Can someone help me with proofs of the following statements?...?

Question

Most helpful person gets best answer:

1. Pythagorean Theorem (a simple proof... please)

2. The bisectors of the angles of a triangle intersect in a point that is equidistant from the 3 sides of the triangle.

3. A dilation maps any triangle to a similar triangle.

4. If a quadrilateral is inscribed in a circle, then the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals.

Help GREATLY appreciated...

Just one proof is fine also...

Answer 1

Use the fact that in a right triangle if an altitude is drawn from the 90 degree angle vertex to the hypotenuse, then the leg^2 = adjacent segment *hypotenuse.
So the altitude divides the hypotnuse (c) into two parts c-x and x.
The above theorem says a^2 = (c-x)c = c^2-cx.
The same theorem states that b^2 =xc
So we have a^2+b^2 = c^2 -cx +cx

Draw two bisectors which intersect at D. Draw perpendiculars from D to the three sides. You wil find that congruent triangles are formed by AAS and thus the perpendicular distances are all the same.

The image will be identical in shape to the preimage and its sides will be proportional according to the dilation factor. Hence, the image and preimage are similar because their sides are proportional.

This is Ptolemy's Theorem. The proof requires auxiliary lines to be drawn extending one side CD of the quad outside the circle to a point P such that the angle DAP = angle BAC.
Then use the fact that opposite angles of a cyclic quad are supplementary, show angle ABC = angle ADP.
Youcan then show that triangle similar triangle DAP and hence DP = (AD*BC)/AB . You can also show CP = AC*BD/AB. Now use CP=CD+DP to arrive at your final result AC*BD=AB*CD+AD+BC