triangle and its circumcircle

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triangle and its circumcircle

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similar triangles

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Source: 2003 British Mathematical Olympiad, Round 2

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#1 h Mar 26, 2012, 4:08 PM • 2 Y

Y by Adventure10, Mango247

Let ABC be a triangle and let D be a point on AB such that
4AD = AB. The half-line l is drawn on the same side of AB as C,
starting from D and making an angle of X with DA where /_X = /_ ACB.
If the circumcircle of ABC meets the half-line l at P, show that
PB = 2PD.

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#2 h Mar 26, 2012, 5:27 PM • 2 Y

Y by Adventure10, Mango247

Obviously, $DP$ is parallel to the tangent at $A$ to the circle $(ABC)$ , hence, if $O$ is its circumcenter, $DP\bot OA$ , hence $\angle APD=\angle ABP$ , i.e. $\triangle ADP\sim\triangle APB$ and $AP^2=AD\cdot AB=\frac{AB^2}{4}\implies AB=2\cdot AP (*)$ ; from the similar triangles we get also $\frac{AP}{AB}=\frac{DP}{BP}$ and, with $(*)$ we get the desired relation.

Best regards,
sunken rock

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