Difference between revisions of "1964 AHSME Problems/Problem 39"
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− | Now, <math>\angle | + | Now, <math>\angle AA'C > \angle B \ge \angle C</math>, hence we have <math>AC > AA' \implies b > AA'</math> |
− | And, <math>\angle | + | And, <math>\angle BB'C > \angle A \ge \angle C</math>, hence we have <math>BC > BB' \implies a > BB'</math> |
− | And, <math>\angle | + | And, <math>\angle CC'B > \angle A \ge \angle B</math>, hence we have <math>BC > CC' \implies a > CC'</math> |
− | Finally, adding all three inequalities, we have <math>b + a + a > | + | Finally, adding all three inequalities, we have <math>b + a + a > AA' + BB' + CC' \implies AA' + BB' + CC' < \textbf{2a + b}</math> |
==See Also== | ==See Also== |
Latest revision as of 21:34, 7 March 2023
Problem
The magnitudes of the sides of triangle are , , and , as shown, with . Through interior point and the vertices , , , lines are drawn meeting the opposite sides in , , , respectively. Let . Then, for all positions of point , is less than:
Solution
We know that in a , if then , we can use this fact in the different triangles to form inequalities, and then add the inequalities.
In , since , we have by the above argument.
Now, , hence we have
And, , hence we have
And, , hence we have
Finally, adding all three inequalities, we have
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 38 |
Followed by Problem 40 | |
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All AHSME Problems and Solutions |
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