Circle Geo Help (1 Viewer)

[blu boi]

ABCD is a cyclic quadrilateral. The diagonals AC and BD intersect at right angles at X. M is the midpoint of BC, and MX is produced to meet AD at N.

a)show that BM=MX
b)show that <MBX = <MXB
c)Show that MN is perpendicular to AD


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[blu boi]

pls help

 

[Drongoski]

a) since <BXC is 90 deg, B, X, C lie on semi-circle, diameter BC, centre M. .: BM = MC = MX, being radii of this semi-circle.

b) since BM = MX, MBX is an isosceles triangle. So base angles MBX and MXB are equal = @ say.

c) .: <DAC = <DBC = @

and < CXM = 90 - < MXB = 90 - @

But < NXA = < CXM = 90 - @ (vertically opp angles)

.:, in triangle ANX, < ANX = 180 - < NAX - < NXA = 180 - @ - (90 - @) = 90

.: MN and AD are perp to each other