perplexus.info :: Just Math : Constant Ratio

Constant Ratio (Posted on 2013-12-17)

A circle has diameter AB and a point P lies on AB between A and B.
A point X, distinct from A and B, lies on the circle.

Prove that tan(∠AXP)/tan(∠XAP) is constant for all values of X.

Submitted by Bractals

Rating: 4.0000 (1 votes)

Solution: (Hide)


Let Q be the foot of the perpendicular from P to AX.
tan(∠AXP) tan(∠QXP) |QP|/|QX| |QA| ----------- = ----------- = ----------- = ------ tan(∠XAP) tan(∠QAP) |QP|/|QA| |QX| |AP| = ------ |PB|
QED

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Possible approach	broll	2013-12-17 22:42:21
Solution	Harry	2013-12-17 17:17:20

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