let triangle ABC be a triangle in which all three interior
angles are acute and let A'B'C' be the orthic triangle.
a.) Prove that the altitudes of triangle ABC are the angle
bisectors of triangle A'B'C'.
b.) Prove the orthocenter of triangle ABC is the incenter of
traingle A'B'C'.
c.) Prove that A is the A' -excenter of triangle A'B'C'.
Prove the following using the triangle inequality:
Given a convex quadrilateral, prove that the point determined by
the intersection of the diagonals is the minimum distance point for
the quadrilateral - that is, the point from which the sum of the
distances of the vertices is minimal.
PROVE THE FOLLOWING RESULT: Using a compass and a straightedge,
construct a triangle ABC such that side BC is of length 2, the
circumradius is of length 3/2 and the median AA' is of length 2 and
is the midpoint of BC.
A triangle ABC has sides AB=50 and AC=10. D is mid-point of AB
and E is mid-point of AC. Angle bisector AG from vertex A meets
side BC at G and divides it in 5:1 ratio. So BG=5 times GC. AG cuts
ED at F. Find the ratio of the areas of the trapeziods FDBG to
FGCE.
Prove Proposition 3.22(SSS Criterion for Congruence). Given
triangle ABC and triangle DEF. If AB is congruent to DE, BC is
congruent to EF, and AC is congruent to DF, then triangle ABC is
congruent to triangle DEF.
(Hint:Use three congruence axioms to reduce to the case where
A=D, C=F, and points B and E are on opposite sides of line AC.)
1. Treacherous Triangle Trickery. Consider a charge distribution
consisting of an equilateral triangle with a point charge q fixed
at each of its vertices. Let d be the distance between the center
of the triangle and each vertex, let the triangle’s center be at
the origin, and let one of its vertices lie on the x-axis at the
point x = −d.
1.1. Compute the electric field at the center of the triangle by
explicitly computing the sum of the...
C. Prove the following claim, using proof by induction. Show
your work.
Let d be the day you were born plus 7 (e.g., if you
were born on March 24, d = 24 + 7). If a =
2d + 1 and b = d + 1, then
an – b is divisible by d for all
natural numbers n.