Difference between revisions of "1978 USAMO Problems/Problem 4"
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== Solution == | == Solution == | ||
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+ | (a) Let <math>ABCD</math> be the said tetrahedron, and let the inscribed sphere of <math>ABCD</math> touch the faces at <math>W, X, Y, Z</math>. Then, <math>OW, OX, OY, OZ</math> are normals to the respective faces. We know that the angle between any two normals is equal, so we have <math>|OW|=|OX|=|OY|=|OZ|</math> at equal angles. Now, since <cmath>WX=2\cdot OW\sin\frac{\angle{WOX}}{2},</cmath> and similar for the other sides, we have that <math>WXYZ</math> is a regular tetrahedron. Now, the faces of <math>ABCD</math> are the tangent planes at <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</math>. Then, consider a <math>120^{\circ}</math> rotation about <math>OW</math>. The rotation sends <math>X\mapsto Y</math>, <math>Y\mapsto Z</math>, and <math>Z\mapsto X</math>. Thus we have <math>AB=AC</math>, <math>AC=AD</math>, <math>AD=AB</math>, and <math>BC=CD=DB</math>. Performing rotations about the other axes yields that <math>ABCD</math> has equal edges, so it is regular. | ||
+ | |||
+ | (b) Consider the normals <math>OW, OX, OY, OZ</math>. We can perform a transformation in which we slightly shift <math>X</math>, <math>Y</math>, and <math>Z</math> closer such that <math>\triangle XYZ</math> is still equilateral, and such that all angles between every pair of normals is less that <math>180^{\circ}</math>. Then, shift <math>W</math> such that <math>WX=WY=XY</math>. Then five of the distances are equal but the sixth is not. | ||
+ | |||
+ | ~ tc1729 | ||
== See Also == | == See Also == |
Revision as of 19:23, 14 April 2015
Problem
(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.
(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?
Solution
(a) Let be the said tetrahedron, and let the inscribed sphere of
touch the faces at
. Then,
are normals to the respective faces. We know that the angle between any two normals is equal, so we have
at equal angles. Now, since
and similar for the other sides, we have that
is a regular tetrahedron. Now, the faces of
are the tangent planes at
,
,
, and
. Then, consider a
rotation about
. The rotation sends
,
, and
. Thus we have
,
,
, and
. Performing rotations about the other axes yields that
has equal edges, so it is regular.
(b) Consider the normals . We can perform a transformation in which we slightly shift
,
, and
closer such that
is still equilateral, and such that all angles between every pair of normals is less that
. Then, shift
such that
. Then five of the distances are equal but the sixth is not.
~ tc1729
See Also
1978 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.