2 High School Construction Problems

[engineerfeelingstupid]

Hi, I'm an engineer and my nephew, who is just learning about symmetry, rotation, etc. got a set of construction problems to do at home. We did all the mandatory ones, but two from the extra credit section just drove me insane. These are the texts of the problems, verbatim:

1. Construct an equilateral triangle if one of its vertices is a given point in the plane, while the other two are points on two given lines.

2. Construct a right triangle from a given hypotenuse and the median to one of the legs.

I'm usually good at geometry, being an engineer and having studied higher math, but for some reason I just can't see the solution. I have gone through half a notebook in two days. My OCD is driving me completely insane.

Please help :|

 

[Ishuda]

Hi, I'm an engineer and my nephew, who is just learning about symmetry, rotation, etc. got a set of construction problems to do at home. We did all the mandatory ones, but two from the extra credit section just drove me insane. These are the texts of the problems, verbatim:

1. Construct an equilateral triangle if one of its vertices is a given point in the plane, while the other two are points on two given lines.

2. Construct a right triangle from a given hypotenuse and the median to one of the legs.

I'm usually good at geometry, being an engineer and having studied higher math, but for some reason I just can't see the solution. I have gone through half a notebook in two days. My OCD is driving me completely insane.

Please help :|

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

Hint for (1). An equilateral triangle has all vertices at the same distance from one another. Points on a circle are all the same distance from the center (of a given point in the plane).

 

[engineerfeelingstupid]

Well, for the first problem I tried to figure out whether the solution would vary based on whether or not the lines are parallel, but the I realized that it would only help if the point is equidistant from both of them. I thought about the circle and using its radius to guarantee the length of the sides from the point in the plane is equal, but if neither of the other two points is defined, how would I know whether the intersects of the circle are the vertices of an equilateral triangle. There is no indication as to at what angle the two lines intersect, so I can't use that angle to determine any other. If I don't have another vertex, I can't figure out a way to determine the precise length of the side, so that I can use a rotation. I've analyzed every possible position of the completed triangle in relation to the two lines, and also it's significant points, but I've had no success in finding a method to precisely define an algorithm that would guarantee an equilateral triangle. As far as I can determine, the circle would only give me a guarantee that the triangle is isosceles.

As for the second problem, I first tried determining properties of a leg's median in relation to the right angle and the hypotenuse in a right triangle, but to no avail. Then I tried extending significant elements of the triangle, so that I would get a shape that includes both given segments in a constant relationship, but still no luck. Then I tried to use both segments as diameters of circles, thus giving me all the points at which the vertex with the right angle would be, hoping that the intersecting points between both circles would give me the right angle, but that also didn't work, since I don't have the angle between the hypotenuse and the median.

I am really sorry to bother you guys/girls, but I have been slamming my head against the desk for the better part of two days, trying to solve high school math problems.