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Straightedge only (Posted on 2015-11-26) Difficulty: 2 of 5
Given a circle with a drawn diameter and an a point K (within the circle, neither on the circumference nor on the diameter line).

Using only a straightedge draw the perpendicular from K to the diameter .

No Solution Yet Submitted by Ady TZIDON    
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Solution Solution Comment 1 of 1

Let the endpoints of the diameter be A and B.

CONSTRUCTION

1) Ray AK intersects the circle again at point C.
2) Ray BK intersects the circle again at point D.
3) Lines AD and BC intersect at point L.
4) Line KL intersects diameter AB at point F.

PROOF

We need to show that /BFK = 90.

Since /ACB and /ADB are subtended by the diameter
AB they are right angles and therefore CKDL is
a cyclic quadrilateral.

   /FKB = /DKL = /DCL = 90 - /DCK
        
        = 90 - /DCA = 90 - /DBA

        = 90 - /KBF

Therefore, /BFK = 90.

QED

  Posted by Bractals on 2015-11-26 12:44:00
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