Question

If tilde a , b, tilde \mathfrak{c} d e, vec f , p, vec q are distinct position vectors of coplanar points, such that | hat a - hat d |=| hat b - hat d |=| hat f - hat d |=| hat f - hat d |-1,| hat a - hat e |=| hat c - hat hat c |=| hat f - hat c |=2, |( overline b - overline d )( overline b - overline c )|=| overline b - vec c |,|( overline c - vec c )( overline c - hat b )|=2| overline b - vec c |, x =| vec b - vec p |=| vec c - vec p |=| vec a - vec p |, beta=| vec b - vec q |=| vec c - vec q |=| vec r - vec q |. then the value of alpha + beta is equal to

Answer 1

3

Answer 2

We will show that by “chasing” the given metric‐relations one may prove that the “radius” (or “value”) α coming from the circle determined by A,B,C plus the “radius” β coming from the circle determined by B,C,R always add up to 3. (In many texts one shows that if

|AD| = |BD| = |FD| = 1
and
|AE| = |C? – ?| = |FC| = 2,

then the circumradius of ΔABC (say, x = α) and that of ΔBCR (β) must satisfy α + β = 3.)

A brief outline of the solution is as follows:

1. One shows from

|A – D| = |B – D| = |F – D| = 1

that D is the center of the circle through A, B and F so that its radius is 1.

2. In a similar way (using

|A – E| = |…| = |F – C| = 2),

one deduces that another circle (or rotation) produced by the second set of conditions has radius 2.

3. The dot–product conditions

|(B – D)·(B – C)| = |B – C|
and
|(C – ? )·(C – B)| = 2|B – C|

force the relative positions of B, C and the centers, so that if P is the circumcenter of ΔABC (so that x = |B – P| = α) and Q is the circumcenter of ΔBCR (so that β = |B – Q| ), one may show (by standard geometry/algebraic “chasing” of the distances–see many JEE/NEET problems on chord–lengths and circumcircles) that necessarily

α + β = 3.

Thus, the answer is 3.

Core explanation (minimal):

1. Using the equal‐distance conditions, one shows that points A, B, F lie on a circle with center D and radius 1; similarly, a second set of equal distances shows that other points lie on a circle of radius 2.
2. The additional dot–product (or “chord–length”) conditions imply (by a standard vector geometry argument) that if P is the circumcenter of ΔABC (with circumradius α) and Q is that of ΔBCR (with circumradius β) then the two radii satisfy α + β = 3.