I actually made one however with the help of an ellipse.
Can the construction be done without using the concept of ellipse? I want another solution since this chapter problem in a book has not yet introduced the concept of ellipse so there maybe a solution.
To anyone asking how I did it with an ellipse here's how: construct a circle with center and end point on mid point and end point of the hypotenuse, respectively. This should act as the median to the hypotenuse as it is half of the hypotenuse (theorem), and the circle act as the locus of the third vertex. Now construct ellipse whose constant lenght is the sum of the base pivoted at the end points of the hypotenuse. The intersection of the circle and ellipse is the vertex that satisfy the condition. The angle between leg should be right by Thales theorem. So there you go, that is my construction.