In an isosceles right-angled triangle ABC (angle of a C-line), points F, D, and G are the midpoints

In an isosceles right-angled triangle ABC (angle of a C-line), points F, D, and G are the midpoints of sides AC, AB, and BC, respectively. Determine the type of quadrilateral CFDG.

Since points F and G are the midpoints of the lateral sides, then the segments CF = CG, and therefore the segment FG is the midline of the triangle ABC, then FG = AB / 2.

Let’s draw the height CD of the ABC triangle, which is also the median of the ABC triangle, then AD = BD = AB / 2.

Triangle ACD is rectangular and isosceles, since the angle CAD = 45, then CD = AD = AB / 2. Segment DF median, bisector and height of triangle ACD, then triangle ADF is rectangular and isosceles, AF = DF = CF. Similarly, ДG = BG = CG.

In the quadrangle CFДG, the diagonals CD and FG are equal and intersect at right angles, and all its sides are equal and parallel, therefore, the quadrangle CFДG is a square.

Answer: Quadrilateral CFDG square.