I am given 3 points (1, 3), (0, 4) and (5, 2) and asked to check that the point of intersection of the perpendicular bisectors is the centre of a circle that passes through all the vertices of the triangle.
I can find the perpendicular bisectors ok but who do prove the intersection?
Am I right in saying all the lines that join the intersection should be of the same length?
Should I find out the lenghts of the perpendicular bisectors?
Best Answer
Let L(1), L(2) and L(3) be the perpendicular bisectors of triangle ABC found.
Normally, three straight lines will NOT meet at the same point (the concurrency point).
Thus, we have to assume that “L(1) and L(2) intersect at one point P”; and “L(2) and L(3) intersect at another point Q”. The question is – “Will P and Q be actually the same point?”
Assuming that has been proved (i.e. Q = P, formally called the circum-center), your are further required to show “Is it the center of the circle circumscribing triangle ABC?”.
About that, your second question is incorrectly asked because you did not mention which points joining that intersection point P are of the same length.
At your level, though it is a proven fact, you are asked to show PA = PB = PC (via distance formula etc).
Furthermore, (1) the correct name of those equaled lengths is called circum-radii; and (2) a perpendicular bisector of a line is infinitely long and thus has no length.