psi20 Posted July 15, 2005 Share Posted July 15, 2005 How do you prove that AD * AC = AB^2? Link to comment Share on other sites More sharing options...
psi20 Posted July 15, 2005 Author Share Posted July 15, 2005 AD is the diameter of the circle, ABC is a right triangle, etc. etc. Link to comment Share on other sites More sharing options...
Primarygun Posted July 16, 2005 Share Posted July 16, 2005 Neither A nor B is equal to 90 degree, so C must be 90 degree. Prove BC^2=ACxCD first, then you can prove the orginal one. There are two methods. (1)Extend BC to E where E is a point on the circumference. Since angle C=90 degree, then it must passes through the centre of the circle. So BC=CE, then you can prove it with Pyth. theorem. You may consider the two similar triangle, the ones like a butterfly. (2)Join BD, angle ABD=90 degree. You can see that ACxCD=BC^2. Either one of the two above leads you the answer. Link to comment Share on other sites More sharing options...
Primarygun Posted July 16, 2005 Share Posted July 16, 2005 By the way, I thought applied maths is the mathematics applying maths into physics,..... So I seldom browse it. I like pure maths. Link to comment Share on other sites More sharing options...
psi20 Posted July 20, 2005 Author Share Posted July 20, 2005 I didn't get that at all. Link to comment Share on other sites More sharing options...
DQW Posted July 20, 2005 Share Posted July 20, 2005 What part didn't you get ? Let's look at PG's solution #2 again : First join BD (at least mentally). Since AD is a diameter, <ABD is a right angle ("angle in a semicircle is a right angle") Hence triangles ACB and BCD are similar (all angles are equal) So, AC/BC = BC/CD (corresponding sides are in the same ratio) Thus, AC*CD = BC^2 But from Pythagoras, BC^2 = AB^2 - AC^2 ... ...plug in and complete... Link to comment Share on other sites More sharing options...
Primarygun Posted July 20, 2005 Share Posted July 20, 2005 Thanks to DQW, my prove is completed . When we prove, we usually do from the backward first. With the given equation, try to split it into some triangles, some other laws. For example, if I splitted and found that I need to prove is XY^2+XZ^2=XY^2, then construct the responsible triangle and then prove angle xyz is equal to 90 degree. Link to comment Share on other sites More sharing options...
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