**EXPLANATION:**

In the figure, we see two planes moving in different directions and at different speeds. The planes send signals towards each other from B and C. When the planes reach A and C, the signals also reach the planes simultaneously.

The lines that connect ABDC points form a Doppler Quadrangle.

Firstly, let’s hide the auxiliary Red and Blue lines by using the button “Auxiliary Lines”. In this way, the quadrangle will be seen more clearly. We see that the side lengths of the quadrangle are different from each other. A parallelogram could have been formed here as well, but the fact that the directions and the speeds of the planes are different lead to the formation of a trapezoid quadrangle.

Now, by using the slider bar, we can observe the movement of the planes and the signals. The flow of events is as follows:

- At the beginning of the event, the planes at B and C send signals towards each other.
- While the signals cover the lines CA (d
_{1}) and BD (d_{3}), the plane on the top covers
CD(d_{2}) and the one on the bottom covers BA (d_{0}).
- When the plane on the top arrives at D, the signal reaches it. When the plane at the bottom arrives at A, the signal reaches it.

**Now we move on to (c+v)(c+v) Mathematics. **

It is true that the signals cover CA (d_{1}) and BD (d_{3}) lines, but this is only true for an observer who is watching the event from the outside. Now, let’s make the auxiliary lines visible and watch the event again.

We see that Red and Blue lines are connected to the reference systems of the planes. Additionally, we also see that Red lines are parallel and equal in length to
d_{4} and that the Blue lines are parallel and equal in length to
d_{5}.

Let’s look at the figure and write the flow of events again:

**Relative to the reference systems of the planes;**

- The Red line which is connected to the plane shows the incoming direction of the signal that comes towards it.
- The Blue line which is connected to the plane shows the outgoing direction of the signal that the plane sends.
- The signal coming towards the plane covers the Red line that belongs to it at c speed. (INCOMING signal speed is always c.). Therefore, the line that determines the travel time of the signal is
“d
_{4}”. d_{4} is the line that represents the distance between the two planes at the moment the signals are sent. The travel time of the signal: t_{Δ}=d_{4}/c
- The signal that is sent from the plane reaches the other plane by following the Blue line that belongs to it. As the travel time of the signal is t
_{Δ}=d_{4}/c, relative to the reference system of the plane, the speed of the signal going towards the other plane on the Blue line is
d_{5}/t_{Δ}=(c±v). d_{5} line represents the distance between the two planes at the moment the signals arrive.

"±" symbol in (c±v):

If
d_{5}<d_{4}, "v" gets a negative value: (c-v)

If
d_{5}>d_{4}, "v" gets a positive value: (c+v)

The distance that the planes will cover within the travel time of the signal

d_{2}=u_{2}.t_{Δ}

d_{0}=u_{1}.t_{Δ}

Now, we can write the speed of the signal relative to the observer on the ground.

The speed of the signal that is sent from the plane at the top to the plane at the bottom:

(c±v_{1})=d_{1}/t_{Δ}

If
d_{4}<d_{1}, "v_{1}" gets a positive value:
(c+v_{1})

If
d_{4}>d_{1}, "v_{1}" gets a negative value:
(c-v_{1})

The speed of the signal that is sent from the plane at the bottom to the plane at the top:

(c±v_{2})=d_{3}/t_{Δ}

If
d_{4}<d_{3}, "v2" gets a positive value: (c+v_{2})

If
d_{4}>d_{3}, "v2" gets a negative value: (c-v_{2})

As you can see, we obtained three different v values.

Relative to the reference systems of the planes and relative to the reference system of the observer.

Consequently, there are the following equations for the Doppler Quadrangle seen in the figure above.

**Travel time of the signal:**

**t**_{Δ}=d_{1}/c

**Side Lengths:**

**d**_{0}=u_{1}.t_{Δ}

**d**_{1}=(c±v_{1}).t_{Δ}

**d**_{2}=u_{2}.t_{Δ}

**d**_{3}=(c±v_{2}).t_{Δ}

**Diagonals:**

**d**_{4}=c.t_{Δ}

**d**_{5}=(c±v).t_{Δ}