How to Install Ubuntu on Mac with High Sierra and Legacy BIOS Booting Windows 10?
Steps Needed to Install Ubuntu after Installing macOS and Windows 10

Here I assumed the following.


The macOS operating system was installed to a partition formatted Mac OS Extended (Journaled). 
Boot Camp was used to install Windows 10.
Windows 10 uses the legacy BIOS boot method.
Windows 10 is version 1803 (i.e. March 2018).
The Desktop Ubuntu version is 18.04.1 LTS.
Ubuntu uses the EFI boot method.
The file ubuntu-18.04.1-desktop-amd64.iso was used to install Ubuntu.



  Note: If you are using APFS or Core Storage, then you will have to make the appropriate minor modifications to these instructions.



Boot to Windows 10.
Download and install gdisk.
Launch the Disk Manager.
Right click on the Windows boot partition and select Shrink volume..., as shown below.


Enter the size needed for Ubuntu. In the example shown below, 100 GB was selected. After entering the amount, select the Shrink button.


The result is shown below. You can exit the Disk Manager. Note: The Disk Manager has updated the Master Boot Record (MBR) partition table, but not the GUID Partition Table (GPT) used by macOS and Ubuntu. This error will be corrected in subsequent steps by using the gdisk command.


Enter the following command in a Administrator Command Prompt window to execute gdisk on your primary internal drive.

gdisk64 \\?\PhysicalDrive0


This will result is output similar to what is shown below.

GPT fdisk (gdisk) version 1.0.4

Partition table scan:
  MBR: hybrid
  BSD: not present
  APM: not present
  GPT: present

Found valid GPT with hybrid MBR; using GPT.

Command (? for help):

Enter the commands needed to copy the size of the Windows boot partition from the Master Boot Record (MBR) partition table to the GUID partition table (GPT). An example is shown below.

Enter the following commands to print out the Master Boot Record (MBR) partition table. 

Command (? for help): r

Recovery/transformation command (? for help): o

Disk size is 1048576000 sectors (500.0 GiB)
MBR disk identifier: 0x3C984A3E
MBR partitions:

Number  Boot  Start Sector   End Sector   Status      Code
   1                     1       411647   primary     0xEE
   2                411648    210126847   primary     0xAF
   3             210126848    211458047   primary     0xAB
   4      *      211458048    843775966   primary     0x07

Recovery/transformation command (? for help):



  Note: You should record the information stored in this table. You will need this information to rebuild the MBR partition table after installing Ubuntu.


The * character shown above indicates partition 4 as the boot partition. The boot partition information needs to be copied to the GUID partition table (GPT). Note: your start and end values will be different from the values of 211458048 and 843775966 used in this example. Also, your boot partition number may be 3 instead of the value of 4 used in this example. Below is an example, where the copy is accomplished be deleting and creating a corrected partition 4.

Recovery/transformation command (? for help): m

Command (? for help): d
Partition number (1-4): 4

Command (? for help): n
Partition number (4-128, default 4): 4
First sector (34-1048575966, default = 211458048) or {+-}size{KMGTP}: 211458048
Last sector (211458048-1048575966, default = 1048575966) or {+-}size{KMGTP}: 843775966
Current type is 'Microsoft basic data'
Hex code or GUID (L to show codes, Enter = 700): 700
Changed type of partition to 'Microsoft basic data'

Command (? for help):


Print out the GPT to verify partition 4 has the correct values, as shown below.

Command (? for help): p
Disk \\?\PhysicalDrive0: 1048576000 sectors, 500.0 GiB
Sector size (logical): 512 bytes
Disk identifier (GUID): A984BAF1-4F22-4E6C-9077-40FEEDF064DB
Partition table holds up to 128 entries
Main partition table begins at sector 2 and ends at sector 33
First usable sector is 34, last usable sector is 1048575966
Partitions will be aligned on 2048-sector boundaries
Total free space is 204802014 sectors (97.7 GiB)

Number  Start (sector)    End (sector)  Size       Code  Name
   1            2048          411647   200.0 MiB   EF00  EFI System
   2          411648       210126847   100.0 GiB   AF00  Apple HFS/HFS+
   3       210126848       211458047   650.0 MiB   AB00  Recovery HD
   4       211458048       843775966   301.5 GiB   0700  Microsoft basic data

Command (? for help):


Write out the changes and quit, as shown below.

Command (? for help): w

Final checks complete. About to write GPT data. THIS WILL OVERWRITE EXISTING
PARTITIONS!!

Do you want to proceed? (Y/N): y
OK; writing new GUID partition table (GPT) to \\?\PhysicalDrive0.
Disk synchronization succeeded! The computer should now use the new
partition table.
The operation has completed successfully.

You can now shutdown Windows 10 and install Ubuntu in the free space.


Adding Partitions for Ubuntu

Below are display snapshots illustrating the partitioning I chose while installing Ubuntu. Basically, I added a new partition for the Ubuntu files and allowed the EFI partition to be default for storing the GRUB boot files. These boot files a stored in a way compatible with the Mac Startup Manager.


When the following display occurs, select Something else, as shown below, then click on the Continue button.


Select free space, as shown below, then click on the + button.


Create swap space. For Use as:, select swap area. For Location for the new partition:, select End of this space. A typical value for swap space is twice the amount of RAM memory installed. Since my Mac has 4 GiB of RAM, the "Size:" field should be 8590 MB. The result is shown below. Next, click on the OK button.


Select free space, as shown below, then click on the + button.


For the Mount point:, select /, as shown below, then click on the OK button.


  Note: The Size: field should automatically default to all the available free space.



The result is shown below. 


  Note: Since End of this space was selected in a previous step, the new partitions will not be created in ascending order. This will cause problems with macOS. In a later step, this problem will be corrected.


Next, click on the Install Now button.


Click on the Continue button shown below to allow the installation to proceed passed the partitioning phase.




Correcting Errors in the GPT and MBR Partition Table

When you installed Ubuntu, the MBR partition table was converted from hybrid to protective. This means most of the values in the MBR partition table were erased. This will prevent Windows from booting. Execute the following steps to correct this problem.


If you have not already done so, boot to Ubuntu and login. You should see a desktop similar to what is shown below.



Press the key combination controloptionT to open a Terminal application window. The result should similar to what is shown below.


Enter the following command to execute gdisk on your primary internal drive. You will be prompted for your login password.

sudo gdisk /dev/sda


This will result in output similar to what is shown below.

GPT fdisk (gdisk) version 1.0.3

Partition table scan:
  MBR: protective
  BSD: not present
  APM: not present
  GPT: present

Found valid GPT with protective MBR; using GPT.

Command (? for help): 


Start by checking to see if the partitions in the GUID Partition Table (GPT) are in ascending order. Enter the command below to print out the GPT.

Command (? for help): p
Disk /dev/sda: 1048576000 sectors, 500.0 GiB
Model: VBOX HARDDISK   
Sector size (logical/physical): 512/512 bytes
Disk identifier (GUID): A984BAF1-4F22-4E6C-9077-40FEEDF064DB
Partition table holds up to 128 entries
Main partition table begins at sector 2 and ends at sector 33
First usable sector is 34, last usable sector is 1048575966
Partitions will be aligned on 2048-sector boundaries
Total free space is 4062 sectors (2.0 MiB)

Number  Start (sector)    End (sector)  Size       Code  Name
   1            2048          411647   200.0 MiB   EF00  EFI System
   2          411648       210126847   100.0 GiB   AF00  Apple HFS/HFS+
   3       210126848       211458047   650.0 MiB   AB00  Recovery HD
   4       211458048       843775966   301.5 GiB   0700  Microsoft basic data
   5      1031798784      1048573951   8.0 GiB     8200  
   6       843776000      1031798783   89.7 GiB    8300  

Command (? for help): 


Partition number 5 should occur after partition 6. Enter the commands given below to swap the values stored in the GPT for these two partitions.

Command (? for help): x

Expert command (? for help): t
Partition number (1-6): 5
New partition number (1-128, default 7): 6

Expert command (? for help):


Enter the command give below to confirm the partitions are now in ascending order.

Expert command (? for help): p
Disk /dev/sda: 1048576000 sectors, 500.0 GiB
Model: VBOX HARDDISK   
Sector size (logical/physical): 512/512 bytes
Disk identifier (GUID): A984BAF1-4F22-4E6C-9077-40FEEDF064DB
Partition table holds up to 128 entries
Main partition table begins at sector 2 and ends at sector 33
First usable sector is 34, last usable sector is 1048575966
Partitions will be aligned on 2048-sector boundaries
Total free space is 4062 sectors (2.0 MiB)

Number  Start (sector)    End (sector)  Size       Code  Name
   1            2048          411647   200.0 MiB   EF00  EFI System
   2          411648       210126847   100.0 GiB   AF00  Apple HFS/HFS+
   3       210126848       211458047   650.0 MiB   AB00  Recovery HD
   4       211458048       843775966   301.5 GiB   0700  Microsoft basic data
   5       843776000      1031798783   89.7 GiB    8300  
   6      1031798784      1048573951   8.0 GiB     8200  


Next, rebuild the MBR partition table. To do this, you will need the values your recorded earlier while booted to Windows. Enter the following.

Command (? for help): r

Recovery/transformation command (? for help): h

WARNING! Hybrid MBRs are flaky and dangerous! If you decide not to use one,
just hit the Enter key at the below prompt and your MBR partition table will
be untouched.

Type from one to three GPT partition numbers, separated by spaces, to be
added to the hybrid MBR, in sequence: 2 3 4
Place EFI GPT (0xEE) partition first in MBR (good for GRUB)? (Y/N): y

Creating entry for GPT partition #2 (MBR partition #2)
Enter an MBR hex code (default AF): af
Set the bootable flag? (Y/N): n

Creating entry for GPT partition #3 (MBR partition #3)
Enter an MBR hex code (default AB): ab
Set the bootable flag? (Y/N): n

Creating entry for GPT partition #4 (MBR partition #4)
Enter an MBR hex code (default 07): 07
Set the bootable flag? (Y/N): y

Recovery/transformation command (? for help): 


If you are prompted with the question given below, enter n. Note: The more experienced individual may want to enter y, then enter 83 when prompted for a MBR hex code.

Unused partition space(s) found. Use one to protect more partitions? (Y/N): 


Print out the MBR partition table to verify the partitions have the correct values, as shown below. The result should be the same as you recorded earlier.

Recovery/transformation command (? for help): o

Disk size is 1048576000 sectors (500.0 GiB)
MBR disk identifier: 0x3C984A3E
MBR partitions:

Number  Boot  Start Sector   End Sector   Status      Code
   1                     1       411647   primary     0xEE
   2                411648    210126847   primary     0xAF
   3             210126848    211458047   primary     0xAB
   4      *      211458048    843775966   primary     0x07

Recovery/transformation command (? for help):


Write out the changes and quit, as shown below.

Recovery/transformation command (? for help): w

Final checks complete. About to write GPT data. THIS WILL OVERWRITE EXISTING
PARTITIONS!!

Do you want to proceed? (Y/N): y
OK; writing new GUID partition table (GPT) to /dev/sda.
Warning: The kernel is still using the old partition table.
The new table will be used at the next reboot or after you
run partprobe(8) or kpartx(8)
The operation has completed successfully.

Quit the Terminal application.
Power off or restart Ubuntu.