Homework # 4

#1

Proof: The angle bisectors of a triangle are concurrent

by

Samuel Obara

The Incenter of a triangle is the point on the interior of the triangle that is equidistant from the three sides. Since a point interior to an angle that is equidistant from the two sides of the angle lies on the angle bisector, then Incenter must be on the angle bisector of each angle of the triangle. Now let us note that triangle AFI and ADI are congruent. This is so because angle IFA and IDA are right angles also BAC has been bisected by the angle bisector hence dividing the angle equally. Which means angle BAI is equal to angle DAI, on the other hand side AI is shared by the two triangle. Hence by side angle side, the two triangle are congruent and therefore by that, we can conclude that since FI is equal to DI. Now using triangle DCI and CEI, note that both share side CI and have angle ACB shared equally among them hence equal angle. Given angle IEC and IDC are right, this make it clear that the two triangles are similar. Now if they are similar, then side DI is equal to side EI. Finally consider triangle BFI and BIE, with same observation, triangle BFI is congruent to BIE, therefore if that is the case, then FI is equal to IE. It is therefore noted that FI = DI = IE hence the angle bisector of a triangle are concurrent. To see the file that created the image, click here.

#2

Proof that if a triangle is inscribed in a circle so that one side of the triangle is the diameterof the circle, then the angle opposite that side is right angle

In the figure below it can be noted that triangle BOA and AOC are isoceles triangle hence angle OBA is equal to angle angle OAB (two sides of raduis are shared as the sides of triangle). On the other hand, angle OAC is equal to OCA. With that in mind, we know that the sum of the angles of triangle add to 180 degrees. Therefore we can state that X+X+Y+Y = 180 which means that 2X +2Y = 180 by dividing by 2 the equation becomes X+Y = 90. Given the angle opposite to the diameter is BAC then we can conclude that this angle is always 90 degrees (X+Y). Lick here for the gsp file.

3. Prove: Let P, Q, R, S be points on the circle C, with P not equal to Q and R not equal to S. Let A be the minor arc of P and Q, and let B be the minor arc of R and S. The central angles of A and B are equal if and only if the chords PQ and RS have the same length.


 

4. Prove: In a regular pentagon the ratio of a diagonal to a side is the golden ratio.

 

1. Each diagonal of the pentagon is parallel to the opposite side. For example the diagonal EB is parallel to the side DC. In other words the quadrilateral
EBCD is an isosceles trapezoid.

2. The angles EBC and CDE are supplementary, and the angles ESC and CDE are equal.

3. The angles SBC and CSB are equal.

4. The triangle SBC is isosceles.

5. The triangles ASB and ABC are similar.

6, Let x = SC = BC, and let y = AS. Then (x+y)/x = x/y.

7. x/y is the golden ratio.

5. Given a triangle ABC, construct a similar triangle A'B'C' so that the area of A'B'C' is twice the area of ABC.

6. Given a line L, a point A, and a line segment BC, construct a circle X so that L is tangent to X, A lies on X, and X has radius of length BC. (Hint: Given
L and r, which circles of radius r have L as a tangent line?)