Difference between revisions of "2018 AIME I Problems/Problem 13"
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First, instead of using angles to find <math>[AI_1I_2]</math>, let's try to find the area of other, more simpler figures, and subtract that from <math>[ABC]</math>. However, to do this, we need to be able to figure out the length of the inradii, and so, we need to find <math>AX</math>. To minimize <math>[AI_1I_2]</math>, intuitively, we should try to minimize the length of <math>AX</math>, since, after using the <math>rs=A</math> formula for the area of a triangle, we'll be able to minimize the inradii lengths, and thus, eventually minimize the area of <math>[AI_1I_2]</math> (I did not come up with a solid proof to back this intuition, if any does, don't hesitate to add a note at the bottom). | First, instead of using angles to find <math>[AI_1I_2]</math>, let's try to find the area of other, more simpler figures, and subtract that from <math>[ABC]</math>. However, to do this, we need to be able to figure out the length of the inradii, and so, we need to find <math>AX</math>. To minimize <math>[AI_1I_2]</math>, intuitively, we should try to minimize the length of <math>AX</math>, since, after using the <math>rs=A</math> formula for the area of a triangle, we'll be able to minimize the inradii lengths, and thus, eventually minimize the area of <math>[AI_1I_2]</math> (I did not come up with a solid proof to back this intuition, if any does, don't hesitate to add a note at the bottom). | ||
− | To minimize <math>AX</math>, let's use Stewart's Theorem. Let <math>AX=d</math>, <math>BX=s</math>, and <math>CX=32-s</math>. After an application of Stewart's Theorem, we will get that <cmath>d=sqrt | + | To minimize <math>AX</math>, let's use Stewart's Theorem. Let <math>AX=d</math>, <math>BX=s</math>, and <math>CX=32-s</math>. After an application of Stewart's Theorem, we will get that <cmath>d=\sqrt{s^2-24s+900}</cmath> To minimize this quadratic, we need to let <math>s=12</math> whereby we conclude that <math>d=6\sqrt{21}</math>. |
From here, draw perpendiculars down from <math>I_1</math> and <math>I_2</math> to <math>AB</math> and <math>AC</math> respectively, and label the foot of these perpendiculars <math>D</math> and <math>E</math> respectively. After, draw the inradii from <math>I_1</math> to <math>BX</math>, and from <math>I_2</math> to <math>CX</math>, and draw in <math>I_1I_2</math>. Label the foot of the inradii to <math>BX</math> and <math>CX</math>, <math>F</math> and <math>G</math>, respectively. From here, we see that to find <math>[AI_1I_2]</math>, we need to find <math>[ABC]</math>, and subtract off the sum of <math>[DBCEI_2I_1]</math>, <math>[ADI_1]</math>, and <math>[AEI_2]</math>. | From here, draw perpendiculars down from <math>I_1</math> and <math>I_2</math> to <math>AB</math> and <math>AC</math> respectively, and label the foot of these perpendiculars <math>D</math> and <math>E</math> respectively. After, draw the inradii from <math>I_1</math> to <math>BX</math>, and from <math>I_2</math> to <math>CX</math>, and draw in <math>I_1I_2</math>. Label the foot of the inradii to <math>BX</math> and <math>CX</math>, <math>F</math> and <math>G</math>, respectively. From here, we see that to find <math>[AI_1I_2]</math>, we need to find <math>[ABC]</math>, and subtract off the sum of <math>[DBCEI_2I_1]</math>, <math>[ADI_1]</math>, and <math>[AEI_2]</math>. |
Revision as of 01:54, 9 November 2020
Contents
Problem
Let have side lengths , , and . Point lies in the interior of , and points and are the incenters of and , respectively. Find the minimum possible area of as varies along .
Solution 1 (Official MAA)
First note that is a constant not depending on , so by it suffices to minimize . Let , , , and . Remark that Applying the Law of Sines to gives Analogously one can derive , and so with equality when , that is, when is the foot of the perpendicular from to . In this case the desired area is . To make this feasible to compute, note that Applying similar logic to and and simplifying yields a final answer of
Solution 2 (A more elegant, but lengthy, approach)
First, instead of using angles to find , let's try to find the area of other, more simpler figures, and subtract that from . However, to do this, we need to be able to figure out the length of the inradii, and so, we need to find . To minimize , intuitively, we should try to minimize the length of , since, after using the formula for the area of a triangle, we'll be able to minimize the inradii lengths, and thus, eventually minimize the area of (I did not come up with a solid proof to back this intuition, if any does, don't hesitate to add a note at the bottom).
To minimize , let's use Stewart's Theorem. Let , , and . After an application of Stewart's Theorem, we will get that To minimize this quadratic, we need to let whereby we conclude that .
From here, draw perpendiculars down from and to and respectively, and label the foot of these perpendiculars and respectively. After, draw the inradii from to , and from to , and draw in . Label the foot of the inradii to and , and , respectively. From here, we see that to find , we need to find , and subtract off the sum of , , and .
can be found by finding the area of two quadrilaterals (+) as well as the area of a trapezoid (). If we let the inradius of be and if we let the inradius of be , we'll find, after an application of basic geometry and careful calculations on paper, that .
The area of two triangles can be found in a similar fashion, however, we must use substitution to solve for as well as . After doing this, we'll get a similar sum in terms of and for the area of those two triangles (I'm not very good at LaTeX, so I can't typeset the area, but the area of the two triangles is equal to (9+3sqrt(21)(r_1)/2 + (7+3sqrt(21)(r_2)/2).
Now we're set. Summing up the area of the Hexagon and the two triangles and simplifying, we get that the formula for is just [ABC]-(35+3sqrt(21)(r_1)/2 -(45+3sqrt(21)(r_2)/2.
Using Heron's formula, . Solving for and using Heron's in and , we get that and . From here, we just have to plug into our above equation and solve. Doing so gets us that the minimum area of .
-Mathislife52
See Also
2018 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.