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Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Geometry Median Problem

Medians

and

of

are perpendicular at point

. Prove that $AB = CG$.

In your diagram,

should appear to be a right angle.

Please try to explain using geometry only terms, like try not to use trigonometry or anything above geometry.

Last edited by Enshrouded_ (2015-08-01 13:40:42)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Geometry Median Problem

hi Enshrouded_

I think this works:

As AGB = 90, then a circle centred on the midpoint, D, of AB will go through G.

(EDIT:  As I read this through, I see I haven't used the circle at all.  Oh well, not to worry.)

Extend AY to E, where AG = GE, and BX to F, where BG = GF.

Join CE and CF.

As G is the median BG = 2GX => GX = XF.

Triangles AGX and CFX are congruent, (SAS) so FC = AG.

Similarly CE = BG.

So triangles AGB, CFG and CEG are congruent (SSS)

So CG = AB.

Bob

Last edited by Bob (2015-08-01 19:56:13)

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