Help of Geometry Problem 39. Triangle, Incircle, Cyclic Quadrilaterals


Suggestions for
Problem 39:

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

DEFINITION 2. Angle is the figure formed by two
rays with a common end point.
Congruent angles are angles
that have the exact same measure (the same number of degrees).
Angle Bisector is a ray that
divides the angle into two congruent adjacent angles.

DEFINITION 3. Perpendiculars are lines or rays or
segments that meet at right angles.

DEFINITION 4. Parallel lines are straight lines which
lie in the same plane and do not intersect however far they
are extended.
PROPOSITION 1. If two lines are parallel, each pair
of alternate interior angles are congruent. Also converse.
PROPOSITION 2. If two
lines are parallel, each pair of corresponding angles are
congruent. Also converse.

DEFINITION 5. 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 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.

DEFINITION 6. Circle is the set of all points in a
plane that are at the same distance from a fixed point
called the center.
Tangent of a circle is a
line that touches the circle at one and only one point no
matter how far produced.
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. Two tangent segments to a circle from an
external point are congruent.

PROPOSITION 13. Isosceles triangle: If two sides of
a triangle are congruent, the angles opposite these sides are
congruent. Also converse.

PROPOSITION 14. A central angle is measured by its
intercepted arc.

PROPOSITION 15. An inscribed angle is measured by
onehalf its intercepted arc.

PROPOSITION 16. A cyclic quadrilateral is a
quadrilateral whose vertices all lie on a single circle.
16.1.
Opposite angles of a cyclic
(inscribed) quadrilateral are supplementary. Also converse.
16.2. A quadrilateral is cyclic if one side subtends
congruent angles at the two opposite vertices. Also
converse.

DEFINITION 7. Similar Triangles
are triangles whose corresponding angles are congruent and whose
corresponding sides are in proportion.
PROPOSITION 17. Triangle Similarity AA. If two
angles of one triangle are congruent to two angles of another
triangle, the two triangles are similar.

