Difference between revisions of "2007 IMO Problems/Problem 4"
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pair A=(0,0),B=(c,0),R=(c/2,-sqrt(25-(c/2)^2)); | pair A=(0,0),B=(c,0),R=(c/2,-sqrt(25-(c/2)^2)); | ||
pair C=intersectionpoints(circle(A,b),circle(B,a))[0]; | pair C=intersectionpoints(circle(A,b),circle(B,a))[0]; | ||
+ | pair K = midpoint(B--C); | ||
pair I=incenter(A,B,C); | pair I=incenter(A,B,C); | ||
dot(A^^B^^C^^R); | dot(A^^B^^C^^R); | ||
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label("$C$",C,N,fontsize(8)); | label("$C$",C,N,fontsize(8)); | ||
label("$R$",R,S,fontsize(8)); | label("$R$",R,S,fontsize(8)); | ||
+ | label("$K$",K,S,fontsize(8)); | ||
+ | |||
</asy> | </asy> | ||
Revision as of 12:29, 24 February 2022
Contents
Problem
In the bisector of intersects the circumcircle again at , the perpendicular bisector of at , and the perpendicular bisector of at . The midpoint of is and the midpoint of is . Prove that the triangles and have the same area.
Diagram
Solution 1 (Efficient)
From a diagram, we have , and similarly . Therefore, we have . Using the area of a triangle formula , we have . By cancelling the sines and the constant, we now have to prove that , or . Draw line perpendicular to BC that intersects BC at . Then because the perpendicular bisectors are congruent, (or alternatively that ) and from that we have by similar triangles. Now the problem is to prove , or .
Since , we have is isosceles . Draw the perpendicular from to , intersecting at . Then for a real . Now, because the perpendicular from the center of a circle to a chord bisects that chord, . Let . Then , proving our claim.
Alternative Solution (Power of a Point)
From , we have . Let the radius of the circumcircle be , then the diameter through is divided by point into lengths of and . By power of point, . Similarly, . Therefore .
Solution by ~KingRavi
Alternate Solution by ~mathdummy
Solution 2
The area of is given by and the area of is . Let , , and . Now and , thus . , so , or . The ratio of the areas is . The two areas are only equal when the ratio is 1, therefore it suffices to show . Let be the center of the circle. Then , and . Using law of sines on we have: so . by law of sines, and , thus 1) . Similarly, law of sines on results in or . Cross multiplying we have or 2) . Dividing 1) by 2) we have
Solution 3
WLOG, let the diameter of be
We see that and from right triangles and
We now look at By the Extended Law of Sines on we get that Similarly,
We now look at By Ptolemy's Theorem, we have which gives us This means that We now seek to relate the lengths computed with the areas.
To do this, we consider the altitude from to This is to find the area of Finding the area of is similar.
We claim that In order to prove this, we will prove that In other words, we wish to prove that This is equivalent to proving that
Note that and Therefore, we get that Thus, In this way, we get that the altidude from to has length Therefore, we see that and so the two areas are equal.
Solution by Ilikeapos
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
2007 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |