Prove that centroid divides the line joining orthocentre and circumcentre in the ratio 2:1.

Asked by yashjain | 6th Dec, 2012, 10:02: PM

Expert Answer:

                                          

                                        

Let O and P be circumcenter and orthocenter respectively. Draw OD and PK perpendicular to BC.

Let AD and OP meet in G.

Now the DAGP and DDGO are equiangular which is clearly due to the fact that OD and PK are parallel lines.

Now OD = OB Cos ÐBOD

               = OB CosA

               = RCosA

Also AP = 2RCosA

So, by similar triangles,

 

So, point G is centroid of the D

Again by the same proposition

 

So, centroid divides line joining circumcenter to orthocenter in ratio 1:2.

Answered by  | 18th Dec, 2012, 10:44: AM

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