MATD02 -Fall 2022 Assignment #1 (Euclidean Geometry)
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MATD02 -Fall 2022
Assignment #1 (Euclidean Geometry)
1. [20marks] Heron’s formula generalizes to cyclic quadrilaterals. Let a, b, c, and d be the lengths of the four sides of a cyclic quadrilateral and P be half of its perimeter. Prove that the area of the quadrilateral is obtained by the following formula:
((P − a)(P − b)(P − c)(P − d)
2. [20marks] Suppose that P is a point on a chord AA' of a given circle T = CO (r).
a) Prove that product of |PA|. |PA′ | is a fixed number (for any chord which passes through P , the product will not change.) This number is called power of the point with respect to the circle and is denoted by 几T (P).
b) The radical axis of two circles T = CO (r) and T ′ = CO′(r′) includes all points P with property that the powers of the point P with respect to both circles are equal. Prove that the radical axis is a line perpendicular to 00′.
3. [ 10marks] Prove that the sum of the medians of a triangle lies between (3/4)P and P where Pis the length of the perimeter of the triangle.
4. [ 10marks] In the Taxicab Geometry, a typical point is a pair of integers P=(x , y). The distance between two points A= (x1 , y1) and B=(x2 , y2) is defined by the following formula:
d(A, B) = |x1 − x2 | + |y1 − y2 |
a) A Taxicab line segment gives the shortest possible path between any two points. Find a formula for the number of Taxicab line segments between two points.
b) Describe the shape of a typical Taxicab circle.
5. [10marks] Suppose x is a root of 
. Is the length of 
is constructible? Explain.
6. [10marks] Describe the symmetry group of a
a) A scalene triangle
b) an isosceles triangle
c) a parabola
d) an ellipse
e) a parallelogram
f) a rectangle
7. [10marks] If H1, H2, H3 are half-turns, prove that H1H2H3 = H3H2H1 .
8. [10marks] Any triangle ∆ABC yields an equilateral triangle ∆PQR if the angles A, B, C are trisected by AQ and AR, BR and BP, CP and CQ by Morley’s Theorem. Prove that three lines PU, QV, RW are concurrent. What values α, β, γ will make the triangle
∆ABC (i) equilateral, (ii) right angled isosceles? Sketch the figure in each case.
α, β, γ are angles the based angle of isosceles triangles constructed on sides QR,
PR, and QP as was explained in the proof of Morley’s Theorem.
2022-12-09