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Proposed Problems 
Problem 29 
Geometry
Help: Problem 29. 
Before beginning the solution of
Problem 29, we need to be aware of the following preliminary
propositions:

DEFINITION 1. Midpoint is the point on a line
segment dividing it into two segments of equal length.

DEFINITION 2. Angle Bisector is a ray that
divides the angle into two congruent adjacent angles.

DEFINITION 3. Triangle is a three side polygon.
Polygon is a closed plane figure with
n
sides. Altitude is the perpendicular line segment from
one vertex to the line that contains the opposite side.

PROPOSITION 1. If two lines are parallel, each pair
of alternate interior angles are congruent. Also converse.

PROPOSITION 2. Two acute angles
are congruent if their sides are respectively perpendicular
to each other.

PROPOSITION 3. The sum of the measures of the three
angles of a triangle is 180.

PROPOSITION 4. The measure of an exterior angle of a
triangle equals the sum of the measures of the two
nonadjacent interior angles.

PROPOSITION 5. The sum of the measures of the acute
angles of a right triangle is 90 (complementary).

PROPOSITION 6. Triangle Congruence S.A.S. If two
sides and the included angle of one triangle are congruent to
the corresponding parts of another, then the triangles are
congruent.

PROPOSITION 7. 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.

PROPOSITION 8. Triangle Congruence S.S.S. If
three sides of one triangle are congruent to the three sides of
a second triangle, then the triangles are congruent.

PROPOSITION 9. Any point on the bisector of
an angle is equidistant from the sides of the angle.

PROPOSITION 10. If a line is
tangent to a circle, it is perpendicular to a radius at the
point of tangency.

PROPOSITION 11. 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.

PROPOSITION 12. The circumcenter O of the triangle
ABC is the concurrence point of the three perpendicular
bisectors of the sides. The circumcenter has the same
distance to the three vertices.

PROPOSITION 13. In the same or congruent circles,
congruent arcs have congruent chords. Also converse.

PROPOSITION 14. Two tangent segments to a circle from an
external point are congruent.

PROPOSITION 15. Isosceles triangle: If two sides of
a triangle are congruent, the angles opposite these sides are
congruent. Also converse.

PROPOSITION 16. For a right triangle with legs a and
b, the hypotenuse c, and the inradius r: a + b = c + 2r.

PROPOSITION 17. Right triangle 454590:
the measure of either leg equals onehalf the measure of the
hypotenuse time
.

PROPOSITION 18. A central angle is measured by its
intercepted arc.

PROPOSITION 19. An inscribed angle is measured by
onehalf its intercepted arc.

PROPOSITION 20. A cyclic quadrilateral is a
quadrilateral whose vertices all lie on a single circle.
Opposite angles of a cyclic
(inscribed) quadrilateral are supplementary. Also converse.


