Prove the following using the triangle inequality: Given a convex quadrilateral, prove that the point determined...

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.

Expert Solution

milcah answered 1 year ago

Prove the following theorem: In a Pasch geometry, a quadrilateral is a convex quadrilateral if and...

Prove the following theorem: In a Pasch geometry, a quadrilateral is a convex quadrilateral if and only if the vertex of each angle is contained in the interior of the opposite angle.

Draw a convex quadrilateral ABCD, where the diagonals intersect at point M. Prove: If ABCD is...

Draw a convex quadrilateral ABCD, where the diagonals intersect at point M. Prove: If ABCD is a parallelogram, then M is the midpoint of each diagonal.

Question: Prove the following: Claim: Consider a triangle ▵ABC and a point D on the interior...

Question: Prove the following: Claim: Consider a triangle ▵ABC and a point D on the interior of segment BC. If σ(▵ABC) = 180, then σ(▵ABD) = σ(▵ACD) = 180. Hint: Use the Split Triangle Theorem and/or the Split Quadrilateral Theorem

PROVE THE FOLLOWING RESULT: Using a compass and a straightedge, construct a triangle ABC such that...

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.

Prove Euclid's Fifth Postulate using Triangle Sum?

Prove that the angle bisectors of a triangle are concurrent using the Pointwise Characterization of an...

Prove that the angle bisectors of a triangle are concurrent using the Pointwise Characterization of an Angle Bisector.

Prove Proposition 3.22(SSS Criterion for Congruence). Given triangle ABC and triangle DEF. If AB is congruent...

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.)

Prove using the principle of mathematical induction: (i) The number of diagonals of a convex polygon...

Prove using the principle of mathematical induction: (i) The number of diagonals of a convex polygon with n vertices is n(n − 3)/2, for n ≥ 4, (ii) 2n < n! for all n > k > 0, discover the value of k before doing induction

For each of the following sets, prove that thay are convex sets or not. Also graph...

For each of the following sets, prove that thay are convex sets or not. Also graph the sets. a) ? 1= {(?1 , ?2 ): ?1 ^2 + ?2^2 ≥ 1} b)?2 = {(?1 ,?2 ): ?1 ^2 + ?2^ 2 = 1} c)?3 = {(?1 , ?2 ): ?1 ^2 + ?2 ^2 ≥ 1}

Considering the illustrations for Concave and Convex mirrors. Prove using geometry that the reflected rays reach...

Considering the illustrations for Concave and Convex mirrors. Prove using geometry that the reflected rays reach the focal point f=R/2 in the limit as the incoming rays approach the principal axis. Hint: Consider the triangle formed by the radius of curvature, principal axis, and reflected ray, and use the law of sines.

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