MathDB

2

Part of 2015 China Northern MO

Problems(2)

angle chasing, tangents at circumcircle of a right triangle

Source: China Northern MO 2015 10.2 CNMO

10/28/2022
It is known that ⊙O\odot O is the circumcircle of △ABC\vartriangle ABC wwith diameter ABAB. The tangents of ⊙O\odot O at points BB and CC intersect at PP . The line perpendicular to PAPA at point AA intersects the extension of BCBC at point DD. Extend DPDP at length PE=PBPE = PB. If ∠ADP=40o\angle ADP = 40^o , find the measure of ∠E\angle E.
geometryanglesdiametercircumcircle
is (AB+BC+CA)/AD^2 irrational?

Source: China Northern MO 2015 grade 11 p2 CNMO

10/28/2022
As shown in figure , a circle of radius 11 passes through vertex AA of △ABC\vartriangle ABC and is tangent to the side BCBC at the point DD , intersect sides ABAB and ACAC at points EE and FF respectively . AlsoEF EF bisects ∠AFD\angle AFD, and ∠ADC=80o\angle ADC = 80^o , Is there a triangle that satisfies the condition, so that AB+BC+CAAD2\frac{AB+BC+CA}{AD^2} is an irrational number, and the irrational number is the root of a quadratic equation with integral coefficients? If it does not exist, please prove it; if it exists, find the quadratic equation that satisfies the condition. https://cdn.artofproblemsolving.com/attachments/b/9/9e3b955b6d6df35832dd0c0a2d1d2a1e1cce94.png
geometryirrational