Draw a line $l$ and two points A and B not lying on l. Make sure that the line $bar(AB)$ is not perpendicular to $l$ . Find the point C on $l$ such that AC = BC?

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Draw a line $l$ and two points A and B not lying on l. Make sure that the line $bar(AB)$ is not perpendicular to $l$ . Find the point C on $l$ such that AC = BC?

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Answer 1 · 2016-10-13T14:17:11

See below.

Explanation:

Given a line

$L_1->p=p_1+lambda_1 vec v_1$

and two points $p_a$ and $p_b$ not pertaining to $L_1$ and not perpendicular to $L_1$

for $p in L_1$ we have

$norm(p-p_a)=norm(p-p_b)$ or

$norm(p_1+lambda_1 vec v_1-p_a)^2=norm(p_1+lambda_1 vec v_1-p_b)^2$ or

$norm(p_1-p_a)^2+2lambda_1 << p_1-p_a, vec v_1 >> +lambda_1^2 norm(vec v_1)^2=norm(p_1-p_b)^2+2lambda_1 << p_1-p_b, vec v_1 >> +lambda_1^2 norm(vec v_1)^2$ so

$norm(p_1-p_a)^2+2lambda_1 << p_1-p_a, vec v_1 >> =norm(p_1-p_b)^2+2lambda_1 << p_1-p_b, vec v_1 >>$ so

$lambda_1^@ = (norm(p_1-p_b)^2-norm(p_1-p_a)^2)/(2<< p_b-p_a, vec v_1 >>)$

The contact point in $L_1$ is

$p_c=p_1+lambda_1^@ vec v_1$

Attached an example with

$p_a = (1,3)$
$p_b = (4,3)$
$p_1 = (1,1)$ and
$vec v_1 = (1,1)$

giving

$p_c = (5/2,5/2)$

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Draw a line $l$ and two points A and B not lying on l. Make sure that the line $bar(AB)$ is not perpendicular to $l$ . Find the point C on $l$ such that AC = BC?

1 Answer

Explanation:

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