# Plot and find intersection points of multiple curves

I tried to plot and find the coordinates of the intersectrions of more than two curves in a the plot. Is there a special way to do this?

Here's my code so yu can visualize the whole thing.

m = 9.1*10^-31;
L = .5 ;
ℏ = 1.055*10^-34;
e = ℏ^2/(2 m) (π/L)^2 (1/(1.602*10^-19));
V1 = 2 e;
V2 = 5 e;
V3 = 8 e;
ν1 = V1/e;
ν2 = V2/e;
ν3 = V3/e;
Plot[
{Sqrt[ν1 - ϵ], Sqrt[ν2 - ϵ], Sqrt[ν3 - ϵ], Sqrt[ϵ] Tan[π/2 Sqrt[ϵ]],
-Sqrt[ϵ] Cot[π/2 Sqrt[ϵ]]},
{ϵ, 0, 10}] • You might be interested in the built-in physical constants, e.g., Entity["PhysicalConstant", "ElectronMass"]["Value"]//QuantityMagnitude, Entity["PhysicalConstant", "ReducedPlanckConstant"]["Value"]//QuantityMagnitude, Entity["PhysicalConstant", "ElementaryCharge"]["Value"]//QuantityMagnitude – Bob Hanlon May 24 '19 at 18:42

plot = Plot[{Sqrt[ν1 - ϵ], Sqrt[ν2 - ϵ], Sqrt[ν3 - ϵ],
Sqrt[ϵ] Tan[π/2 Sqrt[ϵ]], -Sqrt[ϵ] Cot[π/2 Sqrt[ϵ]]}, {ϵ, 0, 10}];

GraphicsMeshMeshInit[]

intersections = GraphicsMeshFindIntersections[plot];
Show[plot, Epilog -> {Red, PointSize[Large], Point@intersections}] • this is great, thanks, but how do i know the coordinates of the points? – Sosa May 24 '19 at 16:58
• @Sosa, intersections is list of coordinates where two lines intersect. – kglr May 24 '19 at 17:06

You can also do this non-graphically:

m = 9.1*10^-31;
L = .5;
hb = 1.055*10^-34;
e = hb^2/(2 m) (\[Pi]/L)^2 (1/(1.602*10^-19));
V1 = 2 e;
V2 = 5 e;
V3 = 8 e;
v1 = V1/e;
v2 = V2/e;
v3 = V3/e;
f1[x_] = Sqrt[v1 - x];
f2[x_] = Sqrt[v2 - x];
f3[x_] = Sqrt[v3 - x];
f4[x_] = Sqrt[x] Tan[\[Pi]/2 Sqrt[x]];
f5[x_] = -Sqrt[x] Cot[\[Pi]/2 Sqrt[x]];
functions = Plot[Evaluate@Through[{f1, f2, f3, f4, f5}[x]], {x, 0, 10}] results = Quiet[Partition[ 