Difference between revisions of "2018 USAMO Problems/Problem 5"
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</asy> | </asy> | ||
− | + | ---- | |
− | <cmath>\angle DEQ + \angle AED + \angle AEP = \angle DAQ + \angle AQD + \angle AEP = 180 - \angle ADC + \angle AEP = 180 - \angle ADC + \angle ABP = \angle ABP + \angle ABC = 180</cmath> | + | <cmath>\begin{align*} |
+ | &\mathrel{\phantom{=}}\angle DEQ+\angle AED+\angle AEP\ | ||
+ | &=\angle DAQ+\angle AQD+\angle AEP\ | ||
+ | &=180-\angle ADC+\angle AEP\ | ||
+ | &=180-\angle ADC+\angle ABP\ | ||
+ | &=\angle ABP+\angle ABC\ | ||
+ | &=180 | ||
+ | \end{align*}</cmath> | ||
so <math>P,E,Q</math> are collinear. Furthermore, note that <math>DQBP</math> is cyclic because: | so <math>P,E,Q</math> are collinear. Furthermore, note that <math>DQBP</math> is cyclic because: | ||
<cmath>\angle EDQ = \angle BAE = BPE.</cmath> | <cmath>\angle EDQ = \angle BAE = BPE.</cmath> |
Latest revision as of 09:46, 27 August 2023
Problem 5
In convex cyclic quadrilateral we know that lines and intersect at lines and intersect at and lines and intersect at Suppose that the circumcircle of intersects line at and , and the circumcircle of intersects line at and , where and are collinear in that order. Prove that if lines and intersect at , then
Solution
so are collinear. Furthermore, note that is cyclic because: Notice that since is the intersection of and , it is the Miquel point of .
Now define as the intersection of and . From Pappus's theorem on that are collinear. It’s a well known property of Miquel points that , so it follows that , as desired. ~AopsUser101
Video Solution by MOP 2024
~r00tsOfUnity