Question

Prove that the internal bisectors of the angles of a triangle are concurrent.

Answer #1

Prove that the bisectors of a triangle are concurrent

Write up a formal proof that the perpendicular bisectors of a
triangle are concurrent, and that the point of concurrency (the
circumcenter) is equidistant from all three vertices

Prove that the intersection of any two perpendicular bisectors of
the sises of a triangle is the center of a circumscribing
circle.

Are the perpendicular bisectors of the sides of a cyclic
quadrilateral concurrent? Sketch an example and make a conjecture.
What would you need to know to prove your conjecture?

Prove: If two angles of a triangle are not congruent, then the
sides opposite those angles are not congruent.

Consider the triangle ABC. Suppose that the perpendicular
bisectors of line segments AB and BC intersect at point X. Prove
that X is on the perpendicular bisector of line segment AC.

Need to Show that for any triangle, the angle bisectors
intersect. Then, show that the intersection point of the medians,
the intersection point of the altitudes, and the intersection point
of the angle bisectors lie on the same line.

if three angles in one triangle are congruent to three angles in
another triangle,then the two triangles are congruent.true or
false,if it is false make a counterexample

Using Euclid's Propositions I.1-I.17, solve the following
questions:
(a) One of the angles of triangle ABC is obtuse. Prove that the
other two are acute.
(b) Prove that the angles at the base of an isoscles triangle
are acute
(c) Prove that every triangle has at least one altitude that is
interior to it.

Prove that if AL, BM, and CN, are proper Cevian lines that are
concurrent at an ordinary point P, then either all three of the
points L, M, and N, lie on triangle ABC or exactly one of them
does. Hint: The three sidelines divide the exterior of triangle ABC
into six regions. consider the possibility that P lies in each of
them separately. Don't forget that one or more of the points L, M,
and N, might be ideal.

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