Wanted: a straight line g, running through P and being perpendicular to h.

But when P has to be a point of the perpendicular bisector, then the distances of P to A and B must be equal! So A and B must lie on a circle with centre P and any radius. We can get suitable points A and B by intersecting the given line h with a (sufficiently big) circle with centre P.

Now maybe you want to start the normal construction of perpendicular bisector - and you are right, this works too! But in this special situation, there is an easier way to the solution: don't we already know, that P has the same distance from A and from B? Then we can use a circle k1 around A running through P and a cricle k2 around B running through P. P is one of the intersection points of k1 and k2. And the second one (let's name it "Q") also has the same distance from A and B. So P and Q are both (different!) points of the perpendicular bisector g of [AB], and so g can be constructed as the straight line running through P and Q!

It's your turn now to construct the wanted perpendicular line g:

- The straight line g that is perpendicular to a given line h and runs through a given

point P lying not on h is called the

- k1 is a circle around P through a point W that may not lie on the same side of h as P
- A and B are the intersection points of k1 and h
- k2 is a circle around A running through P
- k3 is a circle around B running through P
- S and T are the intersection points of k2 and k3
- g is the straight line through S and T

- Why shouldn't W lie "on the same side of h as P"?

Drag W in the drawing above into the "prohibited area"!

Does the construction fail? Why? Always?

- The above condition about the point W is unnecessarily strong.

Can you find a weaker condition, that assures a valid construction, too?