Difference between revisions of "1954 AHSME Problems/Problem 19"
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The same method applies to <math>\angle FED</math> and <math>\angle FDE</math>, which means <math>\triangle DEF</math> is acute - hence our answer is <math>\fbox{D}</math>. | The same method applies to <math>\angle FED</math> and <math>\angle FDE</math>, which means <math>\triangle DEF</math> is acute - hence our answer is <math>\fbox{D}</math>. | ||
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| + | ==See Also== | ||
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| + | {{AHSME 50p box|year=1954|num-b=18|num-a=20}} | ||
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| + | {{MAA Notice}} | ||
Latest revision as of 00:28, 28 February 2020
Problem 19
If the three points of contact of a circle inscribed in a triangle are joined, the angles of the resulting triangle:
Solution
For the sake of clarity, let the outermost triangle be 
with incircle tangency points 
, 
, and 
on 
, 
and 
respectively. Let 
, and 
. Because 
and 
are isosceles, 
and 
. So 
, and since 
, 
is acute.
The same method applies to 
and 
, which means 
is acute - hence our answer is 
.
See Also
| 1954 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
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