MTH 338 Homework #3

Bill OConnell, May 13, 2015

Homework 3 #6, 11, 16, 22 Equivalence to Euclid's 5th postulate.


  • 1. #6 PQ perpendicular to the tangent line at the point of tangency
  • 2. #11 Prove that the perpendicular bisectors of a cyclic quadrilateral are concurrent if and only if the quadrilateral is cyclic.
  • 3. #16 The three excircles of trian...
  • 4. #22 Prove if two circles are orthogonal, then they intersect at exactly two points and their tangents are perpendicular at both of those points
  • 5. Playfair's postulate is logically equivalent to the statement: If a line intersects one of two parallel lines, then it also intersects the other
  • 6. Euclid's fifth postulate is logically equivalent to the statement: If t is a transversal to l and m, l parallel to m and t perpedicular to l, then tis perpendicular to m

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#6 PQ perpendicular to the tangent line at the point of tangency

#6 PQ perpendicular to the tangent line at the point of tangency
Given line l tangent to circle centered at Q at point P. Prove PQ is perpendicular to l. [br]Assume that PQ is not perpendicular to l. Let R be a point on l such that QR is perpendicular to l. The QR<QP since the perpendicular is the shortest distance between Q and l. Since l is tangent at P then that implies that R is outside the circle. Then QR intersects the circle at some point X and QX<QR. Notice that QX=QP then QP<QR this contradicts the assuption that QR<QP. Therefore QP is perpendicular to l.
– Bill OConnell

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