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In a triangle
ABC, median BD is such that angles A and DBC are equal, and mÐ
ADB = 45°, prove that mÐ
A = 30°.
"A great discovery solves a great
problem, but there is a grain of discovery in the solution of
any problem. Your problem may be modest, but if it challenges
your curiosity and brings into play your inventive faculties,
and if you solve it by your own means, you may experience the
tension and enjoy the triumph of discovery. Such expert
experiences at a susceptible age may create a taste for mental
work and leave their imprint on mind and character for a
lifetime." George Polya, 1944
FACTS AND HINTS:
Geometry problem solving is one of the most challenging skills for students to learn. When a
problem requires auxiliary construction, the difficulty of the problem increases drastically, perhaps because deciding which construction to make is an ill-structured problem. By “construction,” we mean adding geometric figures (points, lines, planes) to a problem figure that wasn’t mentioned as "given."
1. Triangle
Proposition: The sum of
the measures of the three angles of a triangle is 180.
Proposition. The measure of an exterior
angle of a triangle equals the sum of the measures of the
two non-adjacent interior angles.
2. Isosceles triangle
Proposition: If two sides of
a triangle are congruent, the angles opposite these sides are
congruent. Also converse.
3. Triangle Congruence A.S.A. If two
angles and the included side of one triangle are congruent to
the corresponding parts of another, then the triangles are
congruent.
4. An equilateral triangle is equiangular: 60-60-60. Also converse.
5. INCENTER of a triangle
Proposition: The bisectors
AD, BF and CE of the angles of a triangle ABC meet in a
point I, which is equidistant from the sides of the
triangle.
The incircle is the inscribed
circle of a triangle. The center of the incircle is called
the incenter, and the radius of the circle is called the
inradius.
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