This exercise involves complex numbers and geometric properties in the complex plane. Let's break it down step by step.

Given Points:

• $A(a)$, $B(b)$, $C(c)$, and $D(d)$ with affixes: $a = 2 - i, \quad b = 1 - 2i, \quad c = -3i, \quad d = \sqrt{2} - \sqrt{2} + i(\sqrt{2} + \sqrt{2})$

Questions Analysis:

1. Calculate distances

   • Use the Euclidean distance formula in the complex plane: $d(A, B) = |a - b| = |(2 - i) - (1 - 2i)|$ Similarly, compute $OA$, $OB$, and $OD$.

2. Determine the nature of triangle $OAB$

   • Use the computed distances to check if $OAB$ is isosceles, right, or equilateral.

3. Show that $A, B, C$ are collinear and verify $AC = 2AB$

   • Check if the vectors $AB$ and $BC$ are proportional.

4. Show that $O, E(1+i), F(1-i), H(2)$ are concyclic

   • Compute their distances from a common center or verify the perpendicular bisectors.

Would you like me to go through the full calculations for each part?

Related Questions:

1. How do you compute distances between complex numbers geometrically?
2. What are the different ways to prove collinearity using complex numbers?
3. How can you determine if four points are concyclic in the complex plane?
4. How does the modulus of a complex number relate to its geometric interpretation?
5. What properties of a triangle can be determined using complex numbers?

Tip:

In complex geometry, distances are computed using the modulus, while collinearity is often checked using proportional differences.