Question

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.

Answer #1

ABC is a right-angled triangle with right angle at A, and AB
> AC. Let D be the midpoint of the side BC, and let L be the
bisector of the right angle at A. Draw a perpendicular line to BC
at D, which meets the line L at point E. Prove that
(a) AD=DE; and
(b) ∠DAE=1/2(∠C−∠B)
Hint: Draw a line from A perpendicular to BC, which meets BC in
the point F

Suppose △ABC and △A'B'C' are triangles such that line AB || to
line A'B', line BC || to line B'C', and line AC || to line A'C'.
Prove that △ABC ~ △A'B'C'.

Given △ABC, extend sides AB and AC to rays AB and AC forming
exterior angles. Let the line rA be the angle bisector ∠BAC, let
line rB be the angle bisector of the exterior angle at B, and let
line rC be the angle bisector of the exterior angle at C.
• Prove that these three rays are concurrent; that is, that they
intersect at a single point. Call this point EA
• Prove that EA is the center of...

Suppose that the incircle of triangle ABC touches AB at Z, BC at
X, and AC at Y . Show that AX, BY , and CZ are concurrent.

Consider a quadrilateral ABCD such that ∠BAD and ∠ADC are
perpendicular, the rays AB and CD are on the same side of the line
AD, and AB ≅ CD.
Prove the following claims on E2, H2, and
S2.
∠ABC ≅ ∠DCB.
the perpendicular bisector of AD is also the perpendicular
bisector of BC.
Hint: Look for symmetries.

Consider a quadrilateral ABCD such that ∠BAD and ∠ADC are
perpendicular, the rays AB and CD are on the same side of the line
AD, and AB ≅ CD. Quadrilaterals with these properties are called
Khayyam quadrilaterals
Prove the following claims on E2, H2, and
S2.
∠ABC ≅ ∠DCB.
the perpendicular bisector of AD is also the perpendicular
bisector of BC.
Hint: Look for symmetries.

If in triangle ABC and Triangle XYZ we have AB = XY, AC = XZ,
but m<A > m<X, then BC > YZ. Conversely, if BC > YZ
then m<A > m<X.

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.

Triangle ABC is a right angle triangle in which
∠B = 90 degree, AB = 5 units , BC = 12 units. CD
and AE are the angle bisectors of ∠C and ∠A
respectively which intersects each other at point I. Find the area
of the triangle DIE.

In triangle ABC , let the bisectors of angle b meet AC at D and
let the bisect of angle C meet at AB at E. Show that if BD is
congruent to CE then angle B is congruent to angle C.

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