2004 Paper 1 Q6

Year: 2004
Paper: 1
Question Number: 6

Course: LFM Pure
Section: Coordinate Geometry

Difficulty: 1484.0 Banger: 1500.0
coordinate geometry centroid medians perpendicularity orthocentre triangle concurrent lines midpoint

Problem

The three points \(A\), \(B\) and \(C\) have coordinates \(\l p_1 \, , \; q_1 \r\), \(\l p_2 \, , \; q_2 \r\) and \(\l p_3 \, , \; q_3 \r\,\), respectively. Find the point of intersection of the line joining \(A\) to the midpoint of \(BC\), and the line joining~\(B\) to the midpoint of \(AC\). Verify that this point lies on the line joining \(C\) to the midpoint of~\(AB\). The point \(H\) has coordinates \(\l p_1 + p_2 + p_3 \, , \; q_1 + q_2 + q_3 \r\,\). Show that if the line \(AH\) intersects the line \(BC\) at right angles, then \(p_2^2 + q_2^2 = p_3^2 + q_3^2\,\), and write down a similar result if the line \(BH\) intersects the line \(AC\) at right angles. Deduce that if \(AH\) is perpendicular to \(BC\) and also \(BH\) is perpendicular to \(AC\), then \(CH\) is perpendicular to \(AB\).

No solution available for this problem.

Rating Information

Difficulty Rating: 1484.0

Difficulty Comparisons: 1

Banger Rating: 1500.0

Banger Comparisons: 0

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Problem source
The three points $A$, $B$ and $C$ have coordinates 
$\l p_1 \, , \; q_1 \r$, $\l p_2 \, , \; q_2 \r$ and 
$\l p_3 \, , \; q_3 \r\,$, respectively. 
Find the point of intersection of the line joining 
$A$ to the midpoint of $BC$, 
and the line joining~$B$ to the midpoint of $AC$. 
Verify that this point lies 
on the line joining $C$ to the midpoint of~$AB$.

The point $H$ has coordinates 
$\l p_1 + p_2 + p_3 \, , \; q_1 + q_2 + q_3 \r\,$. 
Show that if the line $AH$ intersects the line $BC$ at right angles, 
then $p_2^2 + q_2^2 = p_3^2 + q_3^2\,$, 
and write down  a similar result 
if the line $BH$ intersects the line $AC$ at right angles. 

Deduce that if $AH$ is perpendicular to $BC$ and 
also $BH$ is perpendicular to $AC$, then $CH$ is perpendicular to $AB$.
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