Math Is Fun Forum

Let ABCDEFGH be a cube of side length 5, as shown. Let P and Q be points on \overline{AB} and \overline{AE}, respectively, such that AP = 2 and AQ = 1. The plane through C, P, and Q intersects \overline{DH} at R. Find DR.
http://cache.artofproblemsolving.com/asyforum/1/1/8/118d23c1a4792612d4a97f539003b085e17242f2.png
agg how to do this

We are given a cube of side length 1. We then slice a pyramid off each corner, as shown, so that every side length of the remaining polyhedron has the same length. Let A, P, Q, and R be the vertices shown.

Cache.artofproblemsolving.Com/asyforum/8/1/a/81ad9bccc16be74ecf4a456247ceaf4290418ffc.png

import three;unitsize(1 cm);currentprojection=perspective(6,3,2);triple A, B, C, D, E, F, G, H;A = (1,1,0);B = (1,0,0);C = (0...

(a) Let x = AP, y = AQ, and z = AR. Prove that x = y = z.

(b) Find x.

(c) Find the volume of the remaining polyhedron.

wow this ones killing me

Let \overline{PA} and \overline{PC} be tangents from P to a circle. Let B and D be points on the circle such that B, D, and P are collinear. Prove that AB \cdot CD = BC \cdot DA.

This is driving me CRAZY and the moment.
several swearing smileys should make me feel better.

Triangle ABC, inscribed in a circle, has AB = 15 and BC = 25. A tangent to the circle is drawn at B, and a line through A parallel to this tangent intersects \overline{BC} at D. Find DC.
This is weird. I tried stuff I know 'bout circles and triangles...(which is pretty close to nil)...but I can't get it.
Thanks!
:D:D