Difference between revisions of "1967 IMO Problems/Problem 4"
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− | Let | + | Let <math>A_0B_0C_0</math> and <math>A_1B_1C_1</math> be any two acute-angled triangles. Consider all triangles <math>ABC</math> that are similar to <math>\triangle A_1B_1C_1</math> (so that vertices <math>A_1</math>, <math>B_1</math>, <math>C_1</math> correspond to vertices <math>A</math>, <math>B</math>, <math>C</math>, respectively) and circumscribed about triangle <math>A_0B_0C_0</math> (where <math>A_0</math> lies on <math>BC</math>, <math>B_0</math> on <math>CA</math>, and <math>AC_0</math> on <math>AB</math>). Of all such possible triangles, determine the one with maximum area, and construct it. |
− | triangles ABC that are similar to | + | |
− | to vertices A | + | |
− | + | ==Solution== | |
− | possible triangles, determine the one with maximum area, and construct it. | + | We construct a point <math>P</math> inside <math>A_0B_0C_0</math> s.t. <math>\angle X_0PY_0=\pi-\angle X_1Z_1Y_1</math>, where <math>X,Y,Z</math> are a permutation of <math>A,B,C</math>. Now construct the three circles <math>\mathcal C_A=(B_0PC_0),\mathcal C_B=(C_0PA_0),\mathcal C_C=(A_0PB_0)</math>. We obtain any of the triangles <math>ABC</math> circumscribed to <math>A_0B_0C_0</math> and similar to <math>A_1B_1C_1</math> by selecting <math>A</math> on <math>\mathcal C_A</math>, then taking <math>B= AB_0\cap \mathcal C_C</math>, and then <math>B=CA_0\cap\mathcal C_B</math> (a quick angle chase shows that <math>B,C_0,A</math> are also colinear). |
+ | |||
+ | We now want to maximize <math>BC</math>. Clearly, <math>PBC</math> always has the same shape (i.e. all triangles <math>PBC</math> are similar), so we actually want to maximize <math>PB</math>. This happens when <math>PB</math> is the diameter of <math>\mathcal C_B</math>. Then <math>PA_0\perp BC</math>, so <math>PC</math> will also be the diameter of <math>\mathcal C_C</math>. In the same way we show that <math>PA</math> is the diameter of <math>\mathcal C_A</math>, so everything is maximized, as we wanted. | ||
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+ | This solution was posted and copyrighted by grobber. The thread can be found here: [https://aops.com/community/p139266] | ||
+ | |||
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+ | == See Also == | ||
+ | {{IMO box|year=1967|num-b=3|num-a=5}} | ||
+ | [[Category:Olympiad Geometry Problems]] | ||
+ | [[Category:Geometric Construction Problems]] |
Revision as of 13:15, 29 January 2021
Let and be any two acute-angled triangles. Consider all triangles that are similar to (so that vertices , , correspond to vertices , , , respectively) and circumscribed about triangle (where lies on , on , and on ). Of all such possible triangles, determine the one with maximum area, and construct it.
Solution
We construct a point inside s.t. , where are a permutation of . Now construct the three circles . We obtain any of the triangles circumscribed to and similar to by selecting on , then taking , and then (a quick angle chase shows that are also colinear).
We now want to maximize . Clearly, always has the same shape (i.e. all triangles are similar), so we actually want to maximize . This happens when is the diameter of . Then , so will also be the diameter of . In the same way we show that is the diameter of , so everything is maximized, as we wanted.
This solution was posted and copyrighted by grobber. The thread can be found here: [1]
See Also
1967 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |