The perpendicular line in a point of another line

### Problem:

Given: a straight line h and a point M on h.
Wanted: a straight line g, running through M and being perpendicular to h.

### Solution:

The following window shows the start situation. And a possible solution is suggested. Maybe, the point P does not yet have the exactly correct position, but this can easily be repaired - by you!
You can always return to the starting state of the drawing: just select the tool "Reset drawing" from the context menu of the drawing. You get the drawing's context menu with a left mouse button click at any empty spot of the drawing. "Reset drawing" will reload the original state of the construction.
But first of all you should try to understand why the line (MP) is not yet the solution of the problem. The previous thoughts about the perpendicular bisector do not lead us any further here, because there is no segment [AB]! But, this can easily constructed: A and B must lie on h at such positions that M becomes the midpoint of [AB]!

How can you construct two such points? All points that have a certain distance from M lie on a circle around M with this distance as radius. The radius may have any value. So just draw a circle with centre M (use the tool "Circle from centre and peripheral point"): first click the centre point M, then click at a free spot of the Drawing window. A new point is created here, and then the circle is drawn. Now intersect this circle with the line h. The resulting intersection points can now play the roles of A and B.

When you now measure the distances from P to A and B, then it is clear why (MP) cannot be the solution to our problem: the distances are not equal! To make them equal, you have to snap P to the perpendicular bisector of the segment [AB]. So you have to construct the perpendicular bisector g like you've learned on the last page, and then snap P to this line!

### Definition:

The straight line g that runs through a point M of another straight line h and is perpendicular to h,
is called the perpendicular line g in the point M of the line h.

### Result:

Given: a straight line h and a point M on h
Wanted: a straight line g that runs through M and is perpendicular to h
Construction text:
• k1 is a circle with centre M running through any point Q <> M
• A and B are the intersection points of k1 and h
• k2 is a circle around A through B
• k3 is a circle around B through A
• S und T area the intersection points of k2 and k3
• g is the straight line through S and T

### For further puzzling:

1. Does the straight line through S and T will always run through M? Why?

2. Could g be defined as the straight line through S and M?
Which question would remain to be cleared?