 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Circle (Posted on 2005-07-28) C is a circle with center O. AB is a chord not passing through O. M is the midpoint of AB. C' is the circle with diameter OM. T is a point on C'. The tangent to C' at T meets C at P. Show that PA˛ + PB˛ = 4 PT˛.

 No Solution Yet Submitted by nilshady Rating: 3.6667 (3 votes) Comments: ( Back to comment list | You must be logged in to post comments.) Ugly solution | Comment 3 of 5 | Set up a Cartesian coordinate system with the origin at O. We choose our axes such that the x-axis is parallel to the chord AB, and our units such that circle C has radius 1.

Our coordinates are then:
O - (0,0)
A - (-cos x, sin x)
B - (cos x, sin x)
M - (0, sin x)
P - (cos y, sin y)
X - (0, ˝ sin x)
where X is the centre of the smaller circle

Then
PA˛ = (cos y + cos x)˛ + (sin y - sin x)˛
PB˛ = (cos y - cos x)˛ + (sin y - sin x)˛
PT˛ = PX˛ - XT˛ = (cos y)˛ + (sin y - ˝ sin x)˛ - OX˛ (since XT and OX are the radii of the smaller circle)
= (cos y)˛ + (sin y - ˝ sin x)˛ - (˝ sin x)˛

Just expand and it's easy to see how the equality holds...

 Posted by Tan Kiat Chuan on 2005-07-31 18:20:31 Please log in:

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